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 .