DOCSLIB.ORG

Matematisk Ordbok För Högskolan

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Matematisk Ordbok För Högskolan Engelsk-Svensk Svensk-Engelsk Matematisk Ordbok fÄor hÄogskolan Stefan LindstrÄom ii Denna bok Äar typsatt med Times New Roman. Detta verk Äar skyddat av upphovsrÄattslagen. Kopiering Äar fÄorbjuden utÄover vad som anges i kopier- ingsavtal. Den som bryter mot lagen om upphovsrÄatt kan ºatalas av allmÄan ºaklagare och dÄomas till bÄoter eller fÄangelse samt bli skyldig att erlÄagga ersÄattning till innehavaren av upphovsrÄatten. LindstrÄom, Stefan B. ([email protected]) Matematisk ordbok fÄor hÄogskolan Copyright c 2004 iii till Eva-Karin FÄorord Jag bÄorjade skriva Matematisk ordbok fÄor hÄogskolan fÄor att min Eva-Karin lÄaste matem- atikkurser pºa engelska och behÄovde en ordbok. Boken Äar alltsºa skapad med ett synnerligen vÄalbestÄamt mºal: att underlÄatta fÄor studenter pºa universitetsnivºa dºa de lÄaser matematikkurser med engelsk kurslitteratur. Urvalet av uppslagsord bestºar dÄarfÄor i matematiska termer, termer hÄamtade ur typiska matematiska tillÄampningar, ord som fºar sÄarskild eller utÄokad betydelse i en matematisk kontext och slutligen vissa \ovanliga" ord som ofta fÄorekommer i lÄarobÄocker i matematik fÄor att sºadana krÄaver ett exakt uttryckssÄatt. I litteraturfÄorteckningen lÄangst bak ¯nns en lista Äover de bÄocker som ligger till grund fÄor uppgifterna i denna ordbok. Figurerna Äar skapade med programmen x¯g och matlab, medan typsÄattningen Äar gjord i LATEX. iv v Noteringar om ordlistans anvÄandande Denna ordlista Äar tÄankt att anvÄandas av svensktalande, som behÄover ÄoversÄatta ord frºan och till engelska. Boken Äar sÄarskilt skriven fÄor att anvÄandas som stÄod vid matematikstudier med engelsksprºakig lÄarobok. DÄarfÄor ¯nns ofta en lite noggrannare beskrivning pºa svenska vid de engelska uppslagsorden, sºa att lÄasaren skall kunna fÄorstºa dem utan att vara bekant med ordets svenska motsvarighet. Varje uppslagsord beskrivs enligt ett sÄarskilt mÄonster. I den engelsk-svenska delen Äar mÄonstret enligt fÄoljande: 1. Det engelska uppslagsordet stºar pºa en ny rad och Äar skrivet i fetstil. Det kan vara ett ord eller ett begrepp, till och med en lÄangre fras. 2. DÄarefter fÄoljer uppslagsordets ordklass, om det gºar att passa in i en sºadan. Ordklassen Äar skriven som en fÄorkortning i kursiv stil. Uppslagsord med samma stavning, men med skilda ordklasser, betraktas som olika. 3. Sedan, med normalt typsnitt, kommer en lista av motsvarande svenska ord. Ibland stºar ett dolktecken efter ett av orden. I sºa fall Äar det ordet en sÄarskilt god ÄoversÄattning i en matematisk text. 4. FÄor vissa ord fÄoljer sedan ett semikolon och en mer eller mindre omfattande beskrivn- ing av ordets betydelse. Denna beskrivning skall aldrig uppfattas som en matematisk de¯nition, utan bara ses som ett stÄod fÄor fÄorstºaelsen av uppslagsordets innebÄord. 5. Slutligen fÄoljer fÄor vissa ord en rad fraser dÄar uppslagsordets anvÄandning exempli¯eras. Varje fras stºar fÄorst pºa engelska i kursiv stil och sedan pºa svenska i normal stil. Den svensk-engelska delen fÄoljer ett liknande mÄonster, men med nºagra skillnader i sprºakval fÄor olika poster. Dessutom Äar noggrannare beskrivningar av uppslagsorden utelÄamnade. 1. Det svenska uppslagsordet stºar pºa en ny rad i fetstil. Det kan vara ett ord eller ett begrepp. 2. DÄarefter fÄoljer uppslagsordets ordklass. Ordklassen Äar skriven i kursiv stil. Uppslagsord med samma stavning, men med olika ordklass betraktas som olika. 3. Sedan kommer en lista av motsvarande engelska ord och begrepp i normal stil. 4. Slutligen fÄoljer fÄor vissa ord en lista av fraser dÄar uppslagsordets anvÄandning exempli- ¯eras. Varje fras stºar fÄorst pºa svenska i kursiv stil och sedan pºa engelska i normal stil. I det svenska sprºaket skriver man samman ord och bildar nya, medan man i engelskan oftare lºater ord stºa sÄarskrivna. Det ¯nns gott om exempel dÄar svenskan har ett ord fÄor att beskriva nºagonting som engelskan beskriver med tvºa eller er, till exempel svenskans decimaltal och engelskans decimal number. I sºadana fall listas \decimal-" som ett ord i ordlistan med den engelska ÄoversÄattningen \decimal". Ordklassen sÄatts dºa att vara pref. vilket Äar en fÄorkortning fÄor pre¯x. I denna ordbok anvÄands en del fÄorkortningar fÄor vanliga ord och fraser. Tabellen nedan visar en sammanstÄallning Äover dessa. Observera att fÄorkortningar som anvÄands i matematiken Äar listade som uppslagsord med de vanliga orden och inte medtagna hÄar. vi FÄorkortning Betydelse adj. adjektiv adv. adverb alt. alternativt bl.a. bland annat, bland andra el. eller fÄork. fÄorkortning ggr gºanger int. interjektion, utrop konj. konjunktion lÄosn. lÄosning motsv. motsvarande m.a.p. med avseende pºa ngn nºagon ngt nºagot prep. preposition pron. pronomen resp. respektive rÄakn. rÄakneord sub. substantiv symb. symbol t.ex. till exempel (UK) brittisk engelska (US) amerikansk engelska v. verb Äav. Äaven frºan Engelska till Svenska 1 abacus acute angle A absolute value function sub. absolutbe- loppsfunktion, absolutvÄardesfunktion; funk- abacus sub. abacusy, kulram, rÄaknebrÄade. tionen f(x) = x . abbreviation sub. fÄorkortning. absorb v. absorbj jera. Abel's Continuity Theorem sub. Abels absorbing state sub. absorberande tillstºand. kontinuitetssats. abstract sub. sammanfattning; Kort sam- Abel's Impossibility Theorem sub. att manfattning av skrift. polynomekvationer av hÄogre grad Äan tre inte abstract adj. abstrakt. har lÄosningar enbart skrivna med de fyra rÄak- abstract algebra sub. abstrakt algebra. nesÄatten och rotutdragning. abstract space sub. abstrakt rum, rum. Abel's Limit Theorem sub. Abels grÄansvÄar- abundance sub. ÄoverÄod. dessats. accelerate v. accelerera; att Äoka farten. Abel's test sub. Abels test. acceleration sub. hastighetsÄokning, accelera- Abel's Theorem sub. Abels kontinuitetssats; tiony. om kontinuitet hos oÄandliga potensserier. acceleration of gravity sub. tyngdaccelera- Abelian adj. abelsk. tionen; Betecknas g. Dess vÄarde Äar olika pºa Abelian group sub. abelsk grupp, kommuta- olika platser pºa jorden. I engelska bÄocker an- tiv grupp; grupp som Äar kommutativ. vÄands ofta g 9,80 m=s2, i svenska ofta ¼ Abelian integral sub. hyperelliptisk integral. g 9,82 m=s2. ability sub. fÄormºaga. acceleration¼ vector sub. accelerationsvek- about prep. kring, runt, om, omkring. tor. above prep. uppºat, ovan, ovanfÄor, Äover. accent sub. primsymbol, accent. abscindere v. avskÄara, skÄara av. accommodate v. tillfredstÄalla, tillgodose. abscissa sub. abskissa, fÄorsta axel, x-axel. accomplish v. prestera, slutfÄora, uppnºa, ºas- absolute adj. absolut; (positivt) avstºand frºan tadkomma. noll alt. origo till ngt. accomplishment sub. bedrift, prestation. absolute pref. absolut- accordingly adv. alltsºa, fÄoljdaktigt, sºaledes. absolute continuity sub. absolut kontinu- account sub. beaktande, rÄakning; take into itet. account, ta hÄansyn till, ta i beaktande; on absolute convergence sub. absolut konver- somebody's account, fÄor nºagons rÄakning. gens; dºa ngt Äar konvergent Äaven om man fÄorst accumulation point sub. ackumula- tar absolutbeloppet fÄor alla termer. tionspunkt, ackumuleringspunkt, hopn- absolute error sub. absolut fel. ingspunkty. absolute geometry sub. absolut geometri; accuracy sub. noggrannhety, precision. euklidisk geometri utan parallellaxiomet. accuracy of approximation sub. approxi- absolutely convergent adj. absolut konver- mationsnoggrannhet. gent; dºa absolutbeloppet fÄor termerna ger accurate adj. exakt, korrekt, noggranny, pre- konvergens. cis, riktig, trÄa®sÄaker; med liten avvikelse. absolute maximum sub. absolut maximum, accurately adv. exakt, noggrant, precist. globalt maximum. achieve v. lyckas, prestera, uppnºa, ºastadkom- absolute minimum sub. absolut minimum, ma. globalt minimum. achievement sub. bedrift, framgºang, presta- absolute term sub. absolut term, konstant tion. term. act v. bete sig. absolute value sub. absolutbelopp, absolut- activate v. aktivera. vÄarde, belopp. active constraint sub. aktiv begrÄansning, ak- absolute value bar sub. absolutvÄardesstreck; tivt bivillkor. symbolen fÄor absolutbelopp x . active set sub. aktiv mangd. j j Ä acute angle sub. spetsig vinkel; vinkel ® dÄar 0 ® < ¼ . · 2 2 acute triangle allowed acute triangle sub. spetsvinklig triangel; tri- af¯ne plane sub. a±nt plan; ett vanligt plan angel dÄar alla vinklar Äar spetsiga. utan oÄandlighetspunkter. acyclic adj. acyklisk; fÄorlopp utan cykliskt be- af¯ne transformation sub. a±n transforma- teende. tion. acyclic graph adj. acyklisk graf; graf som a±rm v. tillstyrka, intyga. inte innehºaller ngn cykel. a¯eld sub. i fel omrºade. far a¯eld, lºangt frºan Ada sub. Ada; ett programmeringssprºak. aktuellt omrºade. adaptive adj. adaptiv; som anpassar sig. against prep. mot. add v. lÄagga ihop, lÄagga samman, adderay, agree v. instÄamma, stÄamma. summera. air resistance sub. luftmotstºand. addend sub. addend; operand i en addition. aleph symb. alef; tecknet . det fÄorsta tecknet nºagon av a och b i ett uttryck a + b. i det hebreiska alfabetet.@ addition sub. additiony, tillÄagg; att lÄagga alga sub. alg. samman. Algal culture sub. algeodling. additional adj. extra, ytterligare. algebra sub. algebra. addition formula sub. additionsformel. algebraic adj. algebraisk. additive adj. additiv. algebraic basis sub. algebraisk bas, vektor- additive inverse sub. additiv invers, negativ; bas. additive inverse of, negativet till. algebraic complement sub. algebraiskt ad hoc adj. ad hoc; anpassat fÄor ett visst Äan- komplement, kofaktor. damºal. algebraic curve sub. algebraisk kurva. adjacency matrix
Load more

Recommended publications
  • Calculus and Differential Equations II
    Calculus and Differential Equations II MATH 250 B Sequences and series Sequences and series Calculus and Differential Equations II Sequences A sequence is an infinite list of numbers, s1; s2;:::; sn;::: , indexed by integers. 1n Example 1: Find the first five terms of s = (−1)n , n 3 n ≥ 1. Example 2: Find a formula for sn, n ≥ 1, given that its first five terms are 0; 2; 6; 14; 30. Some sequences are defined recursively. For instance, sn = 2 sn−1 + 3, n > 1, with s1 = 1. If lim sn = L, where L is a number, we say that the sequence n!1 (sn) converges to L. If such a limit does not exist or if L = ±∞, one says that the sequence diverges. Sequences and series Calculus and Differential Equations II Sequences (continued) 2n Example 3: Does the sequence converge? 5n 1 Yes 2 No n 5 Example 4: Does the sequence + converge? 2 n 1 Yes 2 No sin(2n) Example 5: Does the sequence converge? n Remarks: 1 A convergent sequence is bounded, i.e. one can find two numbers M and N such that M < sn < N, for all n's. 2 If a sequence is bounded and monotone, then it converges. Sequences and series Calculus and Differential Equations II Series A series is a pair of sequences, (Sn) and (un) such that n X Sn = uk : k=1 A geometric series is of the form 2 3 n−1 k−1 Sn = a + ax + ax + ax + ··· + ax ; uk = ax 1 − xn One can show that if x 6= 1, S = a .
    [Show full text]
  • 1 Mean Value Theorem 1 1.1 Applications of the Mean Value Theorem
    Seunghee Ye Ma 8: Week 5 Oct 28 Week 5 Summary In Section 1, we go over the Mean Value Theorem and its applications. In Section 2, we will recap what we have covered so far this term. Topics Page 1 Mean Value Theorem 1 1.1 Applications of the Mean Value Theorem . .1 2 Midterm Review 5 2.1 Proof Techniques . .5 2.2 Sequences . .6 2.3 Series . .7 2.4 Continuity and Differentiability of Functions . .9 1 Mean Value Theorem The Mean Value Theorem is the following result: Theorem 1.1 (Mean Value Theorem). Let f be a continuous function on [a; b], which is differentiable on (a; b). Then, there exists some value c 2 (a; b) such that f(b) − f(a) f 0(c) = b − a Intuitively, the Mean Value Theorem is quite trivial. Say we want to drive to San Francisco, which is 380 miles from Caltech according to Google Map. If we start driving at 8am and arrive at 12pm, we know that we were driving over the speed limit at least once during the drive. This is exactly what the Mean Value Theorem tells us. Since the distance travelled is a continuous function of time, we know that there is a point in time when our speed was ≥ 380=4 >>> speed limit. As we can see from this example, the Mean Value Theorem is usually not a tough theorem to understand. The tricky thing is realizing when you should try to use it. Roughly speaking, we use the Mean Value Theorem when we want to turn the information about a function into information about its derivative, or vice-versa.
    [Show full text]
  • 3.3 Convergence Tests for Infinite Series
    3.3 Convergence Tests for Infinite Series 3.3.1 The integral test We may plot the sequence an in the Cartesian plane, with independent variable n and dependent variable a: n X The sum an can then be represented geometrically as the area of a collection of rectangles with n=1 height an and width 1. This geometric viewpoint suggests that we compare this sum to an integral. If an can be represented as a continuous function of n, for real numbers n, not just integers, and if the m X sequence an is decreasing, then an looks a bit like area under the curve a = a(n). n=1 In particular, m m+2 X Z m+1 X an > an dn > an n=1 n=1 n=2 For example, let us examine the first 10 terms of the harmonic series 10 X 1 1 1 1 1 1 1 1 1 1 = 1 + + + + + + + + + : n 2 3 4 5 6 7 8 9 10 1 1 1 If we draw the curve y = x (or a = n ) we see that 10 11 10 X 1 Z 11 dx X 1 X 1 1 > > = − 1 + : n x n n 11 1 1 2 1 (See Figure 1, copied from Wikipedia) Z 11 dx Now = ln(11) − ln(1) = ln(11) so 1 x 10 X 1 1 1 1 1 1 1 1 1 1 = 1 + + + + + + + + + > ln(11) n 2 3 4 5 6 7 8 9 10 1 and 1 1 1 1 1 1 1 1 1 1 1 + + + + + + + + + < ln(11) + (1 − ): 2 3 4 5 6 7 8 9 10 11 Z dx So we may bound our series, above and below, with some version of the integral : x If we allow the sum to turn into an infinite series, we turn the integral into an improper integral.
    [Show full text]
  • What Is on Today 1 Alternating Series
    MA 124 (Calculus II) Lecture 17: March 28, 2019 Section A3 Professor Jennifer Balakrishnan, [email protected] What is on today 1 Alternating series1 1 Alternating series Briggs-Cochran-Gillett x8:6 pp. 649 - 656 We begin by reviewing the Alternating Series Test: P k+1 Theorem 1 (Alternating Series Test). The alternating series (−1) ak converges if 1. the terms of the series are nonincreasing in magnitude (0 < ak+1 ≤ ak, for k greater than some index N) and 2. limk!1 ak = 0. For series of positive terms, limk!1 ak = 0 does NOT imply convergence. For alter- nating series with nonincreasing terms, limk!1 ak = 0 DOES imply convergence. Example 2 (x8.6 Ex 16, 20, 24). Determine whether the following series converge. P1 (−1)k 1. k=0 k2+10 P1 1 k 2. k=0 − 5 P1 (−1)k 3. k=2 k ln2 k 1 MA 124 (Calculus II) Lecture 17: March 28, 2019 Section A3 Recall that if a series converges to a value S, then the remainder is Rn = S − Sn, where Sn is the sum of the first n terms of the series. An upper bound on the magnitude of the remainder (the absolute error) in an alternating series arises form the following observation: when the terms are nonincreasing in magnitude, the value of the series is always trapped between successive terms of the sequence of partial sums. Thus we have jRnj = jS − Snj ≤ jSn+1 − Snj = an+1: This justifies the following theorem: P1 k+1 Theorem 3 (Remainder in Alternating Series).
    [Show full text]
  • Series: Convergence and Divergence Comparison Tests
    Series: Convergence and Divergence Here is a compilation of what we have done so far (up to the end of October) in terms of convergence and divergence. • Series that we know about: P∞ n Geometric Series: A geometric series is a series of the form n=0 ar . The series converges if |r| < 1 and 1 a1 diverges otherwise . If |r| < 1, the sum of the entire series is 1−r where a is the first term of the series and r is the common ratio. P∞ 1 2 p-Series Test: The series n=1 np converges if p1 and diverges otherwise . P∞ • Nth Term Test for Divergence: If limn→∞ an 6= 0, then the series n=1 an diverges. Note: If limn→∞ an = 0 we know nothing. It is possible that the series converges but it is possible that the series diverges. Comparison Tests: P∞ • Direct Comparison Test: If a series n=1 an has all positive terms, and all of its terms are eventually bigger than those in a series that is known to be divergent, then it is also divergent. The reverse is also true–if all the terms are eventually smaller than those of some convergent series, then the series is convergent. P P P That is, if an, bn and cn are all series with positive terms and an ≤ bn ≤ cn for all n sufficiently large, then P P if cn converges, then bn does as well P P if an diverges, then bn does as well. (This is a good test to use with rational functions.
    [Show full text]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • Math 6B Notes Written by Victoria Kala [email protected] SH 6432U Office Hours: R 12:30 − 1:30Pm Last Updated 5/31/2016
    Math 6B Notes Written by Victoria Kala [email protected] SH 6432u Office Hours: R 12:30 − 1:30pm Last updated 5/31/2016 Green's Theorem We say that a closed curve is positively oriented if it is oriented counterclockwise. It is negatively oriented if it is oriented clockwise. Theorem (Green's Theorem). Let C be a positively oriented, piecewise smooth, simple closed curve in the plane and let D be the region bounded by C. If F(x; y) = (P (x; y);Q(x; y)) with P and Q having continuous partial derivatives on an open region that contains D, then Z Z ZZ @Q @P F · ds = P dx + Qdy = − dA: C C D @x @y In other words, Green's Theorem allows us to change from a complicated line integral over a curve C to a less complicated double integral over the region bounded by C. The Operator r We define the vector differential operator r (called \del") as @ @ @ r = i + j + k: @x @y @z The gradient of a function f is given by @f @f @f rf = i + j + k: @x @y @z The curl of a function F = P i + Qj + Rk is defined as curl F = r × F: Written out more explicitly: i j k @ @ @ curl F = r × F = @x @y @z PQR @ @ @ @ @ @ = @y @z i − @x @z j + @x @y k QR PR PQ @R @Q @P @R @Q @P = − i + − j + − k @y @z @z @x @x @y 1 The divergence of function F = P i + Qj + Rk is defined as @P @Q @R div F = r · F = + + : @x @y @z Notice that the curl of a function is a vector and the divergence of a function is a scalar (just a number).
    [Show full text]
  • MATH 409 Advanced Calculus I Lecture 16: Tests for Convergence (Continued)
    MATH 409 Advanced Calculus I Lecture 16: Tests for convergence (continued). Alternating series. Tests for convergence [Condensation Test] Suppose an is a nonincreasing { } ∞ sequence of positive numbers. Then the series n=1 an ∞ k converges if and only if the “condensed” series Pk=1 2 a2k converges. P [Ratio Test] Let an be a sequence of positive numbers. { } Suppose that the limit r = lim an+1/an exists (finite or n→∞ ∞ infinite). (i) If r < 1, then the series n=1 an converges. ∞ (ii) If r > 1, then the series n=1 an Pdiverges. P [Root Test] Let an be a sequence of positive numbers and { } n r = lim sup √an. n→∞ ∞ (i) If r < 1, then the series n=1 an converges. ∞ (ii) If r > 1, then the series P n=1 an diverges. P Refined ratio tests [Raabe’s Test] Let an be a sequence of positive numbers. { } a Suppose that the limit L = lim n n 1 exists (finite →∞ n an+1 − ∞ or infinite). Then the series n=1 an converges if L > 1, and diverges if L < 1. P Remark. If < L < then an+1/an 1 as n . −∞ ∞ → →∞ [Gauss’s Test] Let an be a sequence of positive numbers. { } an L γn Suppose that =1+ + 1+ε , where L is a constant, an+1 n n ε> 0, and γn is a bounded sequence. Then the series ∞ { } an converges if L > 1, and diverges if L 1. n=1 ≤ P Remark. Under the assumptions of the Gauss Test, n(an/an+1 1) L as n .
    [Show full text]
  • 11.3 the Integral Test, 11.4 Comparison Tests, 11.5 Alternating Series
    Math 115 Calculus II 11.3 The Integral Test, 11.4 Comparison Tests, 11.5 Alternating Series Mikey Chow March 23, 2021 Table of Contents Integral Test Basic and Limit Comparison Tests Alternating Series p-series 1 ( X 1 converges for = np n=1 diverges for 1 X Z 1 an converges if and only if f (x)dx converges. n=1 1 The Integral Test Theorem (The Integral Test) Suppose the sequence an is given by f (n) where f is a continuous, positive, decreasing sequence on [1; 1). Then Example Determine the convergence of the series of the sequence e−1; 2e−2; 3e−3;::: Table of Contents Integral Test Basic and Limit Comparison Tests Alternating Series 1 1 X X 0 ≤ an ≤ bn n=1 n=1 P1 P1 n=1 an n=1 bn P1 P1 n=1 bn n=1 an The (Basic) Comparison Test Theorem (The (Basic) Comaprison Test) Suppose fang and fbng are two sequences such that 0 ≤ an ≤ bn for all n. Then In particular, If converges, then converges also. If diverges, then diverges also. (What does this remind you of?) Remark We can still apply the basic comparison test if we know that a lim n = L n!1 bn where L is a finite number and L 6= 0. 1 1 X X an converges if and only if bn converges. n=1 n=1 The Limit Comparison Test Theorem (The Limit Comaprison Test) Suppose fang and fbng are two sequences such that Then Example P1 k sin2 k Determine the convergence of k=2 k3−1 Example P1 logpn Determine the convergence of n=1 n n Group Problem Solving Instructions: For each pair of series below, determine the convergence of one of them without using a comparison test.
    [Show full text]
  • 1 Convergence Tests
    Lecture 24Section 11.4 Absolute and Conditional Convergence; Alternating Series Jiwen He 1 Convergence Tests Basic Series that Converge or Diverge Basic Series that Converge X Geometric series: xk, if |x| < 1 X 1 p-series: , if p > 1 kp Basic Series that Diverge X Any series ak for which lim ak 6= 0 k→∞ X 1 p-series: , if p ≤ 1 kp Convergence Tests (1) Basic Test for Convergence P Keep in Mind that, if ak 9 0, then the series ak diverges; therefore there is no reason to apply any special convergence test. P k P k k Examples 1. x with |x| ≥ 1 (e.g, (−1) ) diverge since x 9 0. [1ex] k X k X 1 diverges since k → 1 6= 0. [1ex] 1 − diverges since k + 1 k+1 k 1 k −1 ak = 1 − k → e 6= 0. Convergence Tests (2) Comparison Tests Rational terms are most easily handled by basic comparison or limit comparison with p-series P 1/kp Basic Comparison Test 1 X 1 X 1 X k3 converges by comparison with converges 2k3 + 1 k3 k5 + 4k4 + 7 X 1 X 1 X 2 by comparison with converges by comparison with k2 k3 − k2 k3 X 1 X 1 X 1 diverges by comparison with diverges by 3k + 1 3(k + 1) ln(k + 6) X 1 comparison with k + 6 Limit Comparison Test X 1 X 1 X 3k2 + 2k + 1 converges by comparison with . diverges k3 − 1 √ k3 k3 + 1 X 3 X 5 k + 100 by comparison with √ √ converges by comparison with k 2k2 k − 9 k X 5 2k2 Convergence Tests (3) Root Test and Ratio Test The root test is used only if powers are involved.
    [Show full text]
  • Infinite Sequences and Series Were Introduced Briely in a Preview of Calculus in Con- Nection with Zeno’S Paradoxes and the Decimal Representation of Numbers
    11 Infinite Sequences and Series Betelgeuse is a red supergiant star, one of the largest and brightest of the observable stars. In the project on page 783 you are asked to compare the radiation emitted by Betelgeuse with that of other stars. STScI / NASA / ESA / Galaxy / Galaxy Picture Library / Alamy INFINITE SEQUENCES AND SERIES WERE introduced briely in A Preview of Calculus in con- nection with Zeno’s paradoxes and the decimal representation of numbers. Their importance in calculus stems from Newton’s idea of representing functions as sums of ininite series. For instance, in inding areas he often integrated a function by irst expressing it as a series and then integrating each term of the series. We will pursue his idea in Section 11.10 in order to integrate such functions as e2x 2. (Recall that we have previously been unable to do this.) Many of the func- tions that arise in mathematical physics and chemistry, such as Bessel functions, are deined as sums of series, so it is important to be familiar with the basic concepts of convergence of ininite sequences and series. Physicists also use series in another way, as we will see in Section 11.11. In studying ields as diverse as optics, special relativity, and electromagnetism, they analyze phenomena by replacing a function with the irst few terms in the series that represents it. 693 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
    [Show full text]
  • MA 16200 Study Guide - Exam # 3
    Fall 2011 MA 16200 Study Guide - Exam # 3 (1) Sequences; limits of sequences; Limit Laws for Sequences; Squeeze Theorem; monotone sequences; bounded sequences; Monotone Sequence Theorem. X1 Xn X1 (2) Infinite series an; sequence of partial sums sn = ak; the series an converges to s if sn ! s. n=1 k=1 n=1 (3) Special Series: X1 a n−1 2 3 ··· j j (a) Geometric Series: ar = a(1 + r + r + r + ) = − , if r < 1 (converges). n=1 1 r The Geometric Series diverges if jrj ≥ 1. X1 1 ≤ (b) p-Series: p converges when p > 1; diverges when p 1. n=1 n X1 (4) List of Convergence Tests for an. n=1 (0) Divergence Test (1) Integral Test (2) Comparison Test (3) Limit Comparison Test (4) Alternating Series Test (5) Ratio Test (6) Root Test 1 (Useful inequality: ln x < xm, for any m > > 0:37.) e (5) Strategy for Convergence/Divergence of Infinite Series. X Usually consider the form of the series an: (i) If lim an =6 0 or DNE, the use Divergence Test. n!1 X 1 X (ii) If series is a p-Series , use p-Series conclusions; if series is a Geometric series a rn−1, np use Geometric Series conclusions. P (iii) If an looks like a p-Series or Geometric series, use Comparison Test or Limit Comparison Test. (iv) If an involves factorials, use Ratio Test. th (v) If an involves n powers, use Root Test. P (vi) If an is an alternating series, use Alternating Series Test (or use Ratio/Root Test to show absolute or conditional convergence).
    [Show full text]