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Sequence and Series of Real Numbers

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Sequence and Series of Real Numbers SEQUENCE AND SERIES OF REAL NUMBERS Dr. Vimlesh Assistant Professor, Mathematics Sri Ram Swaroop Memorial University Lucknow-deva Road, U.P. Dr. Pragya Mishra Assistant Professor, Mathematics Pt. Deen Dayal Upadhaya Govt. Girl’s P. G. College, Lucknow, U.P. What is a Sequence? In mathematics a sequence is an ordered list , like a set, it contains members called elements or terms of sequence. Most precisely, a sequence of real numbers is defined as a function S: N R, then for each n∈N, S(n) or 푆푛 is a real number. The real numbers 푆1 푆2 ,푆3,……. ,푆푛 are called terms of sequence. A sequence may be written as *푆1 푆2 푆3……. 푆푛} or {푆푛}. For example 푡ℎ The 푛 term of the sequence {-5n} is 푆푛 = −5푛 then the sequence becomes {-5,-10, -15, ……..-5n…….}. 푛 The 푛푡ℎ term of the sequence {푆 } is 푆 = , the sequence is 푛 푛 푛+1 2 3 {1, , , …}. 3 4 A sequence is also given by its recursion formula where 푆1 = 1and 푆푛 = 3푆푛 , then the sequence is {1, 3, 3√3, … …}. INFINITE SERIES A series is , roughly speaking , a description of the operation of adding infinitely many quantities one after the other, to a given starting quantity. An expression of the form 푎1 + 푎2 + ⋯ +푎푛 + ⋯, where each 푎푛 is real numbers , in which each term is followed by another term is known as infinite series of real ∞ 푡ℎ numbers. It is denoted by 푛=1 푎푛 표푟 푎푛 , here 푎푛 is 푛 term of the series. The sum of n terms of series is denoted by 푆푛 , thus 푛 푆푛 = 푎1 + 푎2 + ⋯ + 푎푛 = 푛=1 푎푛. SEQUENCE OF PARTIAL SUM Suppose 푎푛 is infinite series. We define a sequence *푆푛} as follows: 푆1 = 푎1, 푆2 = 푎1 + 푎2, 푆3 = 푎1 + 푎2 + 푎3, - - 푆푛 = 푎1 + 푎2 + 푎3 + ⋯ + 푎푛 and so on The sequence *푆푛} is called a sequence of partial sums of series 푎푛. CONVERGENCE, DIVERGENCE AND OSCILLATION OF A SERIES 1. CONVERGENT : A series 푎푛 is said to be convergent if the sequence *푆푛} of partial sums of series converges to a real number S. i.e. lim 푆푛 = 푆, where S is finite and unique. 푛→∞ 2. DIVERGENT: A series is said to be divergent if the sequence *푆푛} of partial sum diverges to +∞ 표푟 − ∞. i.e. i.e. lim 푆푛 = ±∞. 푛→∞ 3. OSCILLATORY: A series is oscillatory if sequence *푆푛} of partial sum of series oscillates i.e. lim 푆푛 does not tend to unique limit. 푛→∞ For example: consider the series 1 1 1 + + + ⋯ 1.2 2.3 3.4 1 1 1 1 Here 푆 = + + + ⋯ + 푛 1.2 2.3 3.4 푛(푛+1) 1 1 1 1 1 1 1 = 1 − + − + − + ⋯ + − 2 2 3 3 4 푛 푛+1 1 1 푆푛 = 1 − lim 푆푛 = lim 1 − = 1 − 0 = 1 푓푖푛푖푡푒 . 푛+1 푛→∞ 푛→∞ 푛+1 Hence the series converges to 1. Consider the series 푛 2 3 3 = 3 + 3 + 3 + ⋯ 푛 푛 Here 푆푛 = 3 − 1 , lim 푆푛 = lim 3 − 1 = ∞. 푛→∞ 푛→∞ Hence the series is divergent. Example: Consider the series (−ퟏ)풏−ퟏ . 푛−1 • Here 푎푛 = (−1) 푛표푤 푆1 = 푎1 = 1 • 푆2 = 푎1 + 푎2 = 1 − 1 = 0, • 푆3 = 푎1 + 푎2 + 푎3 = 1 − 1 + 1 = 1, 푆4 = 0, 푆5 = 1 … . 푠표 표푛. • Therefore *푆푛+ = 1,0,1,0, … Which Oscillates between 0 and 1.So the series is oscillatory. Elementary Properties of series The alteration of a finite number of terms of a series has no effect on convergence and divergence. 1. If a series converges or has an infinite sums, their sum is unique. 2. Multiplication of the terms of a series by a nonzero constant K doesnot effect the convergence or divergence of a series. ∞ ∞ 퐾푎푛 = 푘 푎푛 푛=1 푛=1 Necessary Condition For Convergence: Theorem: If a series converges, its general term tends toward zero as n becomes infinity i.e. if the series 푎푛 converges , then lim푛→∞푎푛 = 0. But the converse is not true. Proof: let us consider the series 푎푛 . Consider *푆푛} be the sequence of partial sums of series 푎푛 . 푆푛 = 푎1 + 푎2 + ⋯ + 푎푛−1 + 푎푛 푆푛−1 = 푎1 + 푎2 + ⋯ + 푎푛−2 + 푎푛−1 푆푛 − 푆푛−1 = 푎푛 Since the series 푎푛 converges , so *푆푛} converges Let lim 푆푛 = 푠 푡푕푒푛 lim 푆푛−1 = 푠 푛→∞ 푛→∞ Therefore lim 푎푛 = lim 푆푛 − lim 푆푛−1 = 푠 − 푠 = 0 푛→∞ 푛→∞ 푛→∞ Hence lim 푎푛=0 푛→∞ Thus the condition for convergence is necessary but not sufficient. Consider the harmonic series 1 1 1 1 = 1 + + + ⋯ + + ⋯ 푛 2 3 푛 1 1 Here 푎푛= lim푎푛 = lim = 0 푛 푛→∞ 푛→∞ 푛 1 But the series is divergent . (by p-test) 푛 RESULT FOR GEOMETRIC SERIES The geometric series 푟푛 = 1 + 푟 + 푟2 + 푟3 + ⋯ (푟 > 0) is convergent if 푟 < 1 and divergent if 푟 ≥ 1. Examples: 1 1 1 1 The series = + + ⋯ + + ⋯ is convergent. 2푛 2 22 2푛 Here the series is G.P. series where common ratio 푟 = 1/2 < 1. so the above series converges. The series 3푛 = 3 + 32 + 33 + ⋯ is divergent, since 푟 = 3 > 1. POSITIVE TERM SERIES An infinite series whose terms are positive or more generally, a non negative series (a series whose terms are nonnegative) are called positive term series. Theorem: (A test for a positive series) A positive term series 푎푛 is convergent if and only if its sequence *푆푛} of partial sum bounded above. Equivalently, a positive term series 푎푛 converges iff 푆푛 < 푘 ∀푛 ∈ 푁. Remark: Since monotonic sequence either converge or diverge but never oscillate. Therefore positive term series are either converge or diverge. Test for Divergence Theorem: If 푎푛is a positive series such that lim 푎푛 ≠ 0, then the 푎푛 diverges. 푛→∞ 휋 EXAMPLE(1): Test the convergence of series ∞ 푐표푠 . 푛=1 2푛 휋 휋 Here 푎푛 = 푐표푠 , Then lim 푎푛 = lim 푐표푠 = cos0 = 1 ≠ 0. 2푛 푛→∞ 푛→∞ 2푛 So by theorem lim 푎푛 ≠ 0 then the series is divergent. 푛→∞ 1 2 푛 Example (2): consider the series + + ⋯ + + ⋯ 푛 6 2 푛+1 푛 푛 1/2 1 1/2 Here 푎푛 = lim 푎푛 = lim = lim 2 푛+1 푛→∞ 푛→∞ 2푛 1+1/푛 푛→∞ 2 1+1/푛 1 = ≠ 0 2 Hence the series is divergent. SOME COMPARISON TEST FIRST COMPARISON TEST: Theorem: let 푎푛 and 푏푛be two positive term series such that 푎푛≤ 푘 푏푛 ∀푛 ≥ 푚 (k being a fixed positive number and m a fixed positive integer. Then 1. If 푏푛 converges implies 푎푛converges. 2. If 푎푛diverges implies 푏푛diverges. 1 CONVERGENCE OF p-SERIES 푛푝 ퟏ ퟏ ퟏ ퟏ The series ∞ = + + ⋯ + + ⋯ (퐩 > ퟎ) 풏=ퟏ 풏풑 ퟏ풑 ퟐ풑 풏풑 Converges if (p >1) and diverges if 풑 ≤ ퟏ . 2 EXAMPLE consider the series 푒−푛 푒푥 > 푥 ∀푥 > 0\ 2 푒푛 > 푛2 ∀푛 1 1 −푛2 1 2 < 푒 < ∀푛 푒푛 푛2 푛2 1 Since ∞ converges (by p-series test here p=2 >1) 푛=1 푛2 2 Hence by first comparison test 푒−푛 is convergent. ∞ 1 EXAMPLE: Consider series 푛=2 푛2 log 푛 1 1 Since we know that > ∀푛 ≥ 2 푛2 log 푛 푛2 1 Here series ∞ is convergent since p = 2 > 1 푛=1 푛2 So by first comparison test ∞ 1 is convergent. 푛=2 푛2 log 푛 LIMIT FORM TEST Theorem Let 풂풏 and 풃풏 be two positive term series such that 풂 퐥퐢퐦 풏 = 풍 (풍 풊풔 풏풐풏풛풆풓풐 풂풏풅 풇풊풏풊풕풆) 풏→∞ 풃풏 Then 풂풏 and 풃풏 converges and diverges together. i.e. 풃풏converges implies 풂풏 converges. 풃풏 diverges implies 풂풏 diverges. REMARKS: 1. If l = 0 표푟 푙 = ∞ then above test may not hold good. 2. To apply limit test on the series 푎푛, we have to select series 푏푛 called auxillary 푡ℎ series (which is usually p-series) in which the 푛 term of 푏푛 behaves as 푎푛, for large values of n written as 푎푛~푏푛. For large values of n we have 1 1 ~ 푛2 + 1 푛2 1 1 1 ~ = . 푛+ 푛+1 푛+ 푛 2 푛 We also take 푏푛 as 1 푏 = where 푎 푎푛푑 푏 are higher indices of n in denominator and numerator. 푛 푛(푎−푏) 푛 1 1 For example 푎 = then 푏 = = 푛 푛3+ 푛 푛 푛3−1 푛2 푎 And usually lim 푛 = 1. 푛→∞ 푏푛 1 The series converges if 푝 > 1 and diverges if 푝 ≤ 1. 푛푝 EXAMPLES 1 3 5 consider the series + + + ⋯ 1.2.3 2.3.4 3.4.5 The 푛푡ℎ term of the series is 2푛 − 1 푎 = 푛 푛(푛 + 1)(푛 + 2) 1 2 − 푎 = 푛 푛 1 2 푛2(1 + )(1 + ) 푛 푛 1 Let us consider the auxiliary series 푏 = 푛 푛2 2 푎푛 (2푛−1)푛 Then = 푏푛 푛(푛+1)(푛+2) 풂 푛(2푛 − 1) 풏 = 풃풏 (푛 + 1)(푛 + 2) 풂 풏 ퟐ풏 − ퟏ lim 풏 = lim 푛→∞ 풃풏 푛→∞ ퟏ + 풏 풏 + ퟐ 푎 1 2 − 1\푛 푙푖푚 푛 = 푙푖푚 1 푛→∞ 푏푛 푛→∞ 1 + 푛 1 + 2\푛 풂 ퟏ ퟐ − ퟎ lim 풏 = 푛→∞ 풃풏 ퟏ + ퟎ ퟏ + ퟎ = 2 ≠ 0 푓푖푛푖푡푒 1 So 푎 and 푏 converges or diverges together since the series 푏 = converges 푛 푛 푛 푛2 (because p=2>1). Hence 푎푛 converges. ∞ 4 4 Example: Test the convergence of the series 푛=1 푛 + 1 − 푛 − 1 Sol: Here 푛푡ℎ term of the series will be 푛4 + 1 + 푛4 − 1 4 4 푎푛 = 푛 + 1 − 푛 − 1 × 푛4 + 1 + 푛4 − 1 푛4 + 1 − 푛4 − 1 2 푎푛 = ~ 푛4 + 1 + 푛4 − 1 1 1 n2 1 + + n2 1 − n4 n4 1 Consider 푏 = then 푛 푛2 푎 2 lim 푛 = lim 푛→∞ 푏푛 푛→∞ 1 1 1 + + 1 − 푛4 푛4 2 = = 1 ≠ 0(푓푖푛푖푡푒) 2 1 So 푎 and 푏 converges or diverges together .
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  • 1 Notes for Expansions/Series and Differential Equations in the Last
    Notes for Expansions/Series and Differential Equations In the last discussion, we considered perturbation methods for constructing solutions/roots of algebraic equations. Three types of problems were illustrated starting from the simplest: regular (straightforward) expansions, non-uniform expansions requiring modification to the process through inner and outer expansions, and singular perturbations. Before proceeding further, we first more clearly define the various types of expansions of functions of variables. 1. Convergent and Divergent Expansions/Series Consider a series, which is the sum of the terms of a sequence of numbers. Given a sequence {a1,a2,a3,a4,a5,....,an..} , the nth partial sum Sn is the sum of the first n terms of the sequence, that is, n Sn = ak . (1) k =1 A series is convergent if the sequence of its partial sums {S1,S2,S3,....,Sn..} converges. In a more formal language, a series converges if there exists a limit such that for any arbitrarily small positive number ε>0, there is a large integer N such that for all n ≥ N , ^ Sn − ≤ ε . (2) A sequence that is not convergent is said to be divergent. Examples of convergent and divergent series: • The reciprocals of powers of 2 produce a convergent series: 1 1 1 1 1 1 + + + + + + ......... = 2 . (3) 1 2 4 8 16 32 • The reciprocals of positive integers produce a divergent series: 1 1 1 1 1 1 + + + + + + ......... (4) 1 2 3 4 5 6 • Alternating the signs of the reciprocals of positive integers produces a convergent series: 1 1 1 1 1 1 − + − + − + ......... = ln 2 .
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  • Chapter 3: Infinite Series
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  • Approximating the Sum of a Convergent Series
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