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Calculus Convergence and Divergence

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Calculus Convergence and Divergence CALCULUS CONVERGENCE AND DIVERGENCE TEST NAME SERIES CONVERGES DIVERGES ADDITIONAL INFO 1 nth TERM TEST an if lim an =0 One should perform this n 6 n=1 !1 test first for divergence. X 1 n 1 GEOMETRIC anr − if 1 <r<1 if r 1 If convergent, converges − | |≥ a n=1 SERIES TEST to sn = 1 r X − 1 1 P-SERIES TEST if p>1 if p 1 Can be used for compar- np n=1 ison tests. X 1 INTEGRAL f(x) if 1 f(x) dx if 1 f(x) dx f(x) has to be continu- 1 · 1 · TEST n=1 converges. diverges. ous, positive, decreasing X R R on [1, ). 1 1 DIRECT a if 0 a b , and if 0 b a , and For convergence, find a n n n n n COMPARISON n=1 1 1 larger convergent series. X bn converges. bn diverges. TEST n=1 n=1 For divergence, find a P P smaller divergent series. 1 1 1 LIMIT an if bn converges, if bn diverges, If necessary, apply COMPARISON n=1 n=1 n=1 L’Hospital’s Rule. X an an TEST andP lim > 0. andP lim > 0. Inconclusive if n bn n bn a !1 !1 lim n = 0 or . n !1 bn 1 1 n+1 ALTERNATING ( 1) an if an+1 an, and if lim an = 0. To prove convergence − n 6 SERIES TEST n=1 lim an = 0. !1 prove that the sequence X n !1 is decreasing and its limit is zero. 1 an+1 an+1 RATIO TEST an if lim < 1. if lim > 1. The test fails if n a n a n=1 !1 n !1 n an+1 X lim = 1. n a !1 n 1 n n ROOT TEST an if lim an < 1. if lim an > 1. The test fails if n | | n | | n n=1 !1 !1 lim an = 1. X p p n !1 | | p csusm.edu/stemsc xxx @csusm_stemcenter Tel: North: 760-750-4101 South: 760-750-7324 CALCULUS CONVERGENCE AND DIVERGENCE DEFINITION OF CONVERGENCE AND DIVERGENCE 1 An infinite series an = a1 + a2 + a3 + ... is convergent if the sequence sn of partial sums, where n=1 { } P n each partial sum is denoted as sn = an = a1 + a2 + ...+ an, is convergent. n=1 If the sequence sn is divergent, thenP the series is called divergent. { } ABSOLUTELY CONVERGENT CONDITIONALLY CONVERGENT Aseries an is called absolutely convergent if Aseries an is called conditionally convergent the series of the absolute values an is if it is convergent but not absolutely convergent. convergent.P | | P P 1 1 1 1 1 1 1 1 c an = c an (an + bn)= an + bn (an bn)= an bn n=1 · · n=1 n=1 n=1 n=1 n=1 − n=1 − n=1 P P P P P P P P POWER SERIES 1 n 2 3 A power series is a series of the form cnx = c0 + c1x + c2x + c3x + ... where x is a variable and n=0 the cn’s are called the coefficients of theP series. 1 n 2 A series of the form cn(x a) = c0 + c1(x a)+c2(x a) + ... is called a power series in (x a) n=0 − − − − or a power series centeredP at a or a power series about a. 1 n For a given power series cn(x a) there are only three possibilities: n=0 − P (i) The series converges only when x = a. (ii) The series converges for all x. (iii) There is a positive number R such that the series converges if x a <Rand diverges if x a >R. | − | | − | 1 1 =1+x + x2 + x3 + ...= xn x < 1 1 x | | n=0 − X 1 n If the power series cn(x a) has radius of convergence R>0, then the function defined by n=0 − 1 Pn f(x)= cn(x a) is di↵erentiable on the interval (a R, a + R) and n=0 − − P 1 n 1 (i) f 0(x)= ncn(x a) − . n=0 − P 1 (x a)n+1 (ii) f(x)=C + cn −n+1 . n=0 R P csusm.edu/stemsc xxx @csusm_stemcenter Tel: North: 760-750-4101 South: 760-750-7324.
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