Dr. Dharmendra Kumar Yadav Assistant Professor of Mathematics M. L. S. College, Sarisab-Pahi, Madhubani R. K. College, Madhubani Lecture Notes Part-2 Paper-III B. Sc. Maths (H) CONVERGENCY AND DIVERGENCY OF INFINITE SERIES Part-1 Paper-I B. Sc. Maths (H) Summation of Trigonometric Series 01/04/2020 [M.L.S. College] 04/04/2020 [R. K. College] Part-2 Paper-III CONVERGENCY AND DIVERGENCY OF INFINITE SERIES Series: An expression in which the successive terms are formed by some regular law or definite law is called a series or simply the sum of the terms of a sequence is called a series. An expression of the form u1 + u2 + u3+…..un +……. is a series, in which u1, u2,..…un , ….are its terms. If the sum of the infinite series can be obtained it comes under convergent series otherwise under divergent or oscillatory series. Finite and Infinite Series: If a series has finite number of terms, it is called finite otherwise infinite. In finite series, the series terminates at some assigned term. In the infinite series, the series do not terminate. Note: Every finite series is a convergent series. Sum Notation: If {an} be the given series then an or an is denoted for infinite series. n1 Sum of Series: The sum of a finite series is finite because there is a last term in the case of a finite series. The sum of a finite series is the term by term addition. In the infinite series there is no last term so the process of addition will never be completed if we apply term by term addition. Hence in the ordinary case, the sum of an infinite series has no meaning. Therefore, the notion of limit has been employed to find out the sum of an infinite series. Convergent Series (Sum of Convergent Series): An infinite series is said to be convergent when the sum of the first n terms cannot numerically exceed some finite quantity however greater n may be, i.e. Sn ( Sn a1 a 2 a 3 .......... a n ) may tend to a definite finite number S such that when n is made sufficiently large, Sn-S is as small as we please. It may be written lim as SS , (where S a finite number), i.e. to every positive number 0 however n n small a positive integer m such that SSn when n m . lim Working Rule: If S exists and is equal to S a finite and unique quantity, then the n n series is called convergent and S is called its sum, otherwise the series is called divergent. If a series is convergent, its sum is finite. Divergent Series (Sum of Divergent Series): If the sum of an infinite series is greater than a finite quantity, i.e. if the sum is or , then the series is said to be divergent series. Generally, a positive term series which is not convergent is a divergent series. In other words, in an infinite series if lim lim S or S n n n n Then, the series is divergent. If a series is divergent, its sum is infinite. Example: 2n and n2 are divergent series. Oscillating Series (Sum of Oscillating Series): If the sum of an infinite series does not tend to a unique limit as n , finite or infinite, then the series is said to be an oscillating series. Example: ( 1)n1 . Note: If lim S n n is not unique, the series ∑an is known as oscillatory, sometimes called non-convergent series. Positive Term Series: An infinite series in which all the terms from and after some finite number of term are positive, is called a positive term infinite series or positive term series. 1 1 1 Example: 1 2 3 1 ............. 2 4 8 Fundamental Theorem of Infinite Series: The convergence or divergence of an infinite series remains unaffected by removal or addition of a finite number of terms. Note: Regrouping of terms does not affect the nature of the series. Part-1 Paper-I Summation of Trigonometric Series Generally we use two methods to find the summation of the trigonometric series: 1. Method of Differences, and 2. C + iS Method. Method of Differences We express each term of the series as the difference of two terms, so that one of the terms is the same function of the integer r as the other is of the (r-1) or (r+1). If we want to find the sum of the series u1 + u2 + u3 + … + un, We write ur = f(r+1) - f(r) or ur = f(r) - f(r-1). If we can write ur as ur = f(r+1) - f(r), then putting r = 1, 2, 3, … successively, we get u1 = f(2) - f(1) u2 = f(3) - f(2) u3 = f(4) - f(3) ……………… un-1 = f(n) - f(n-1) un = f(n+1) - f(n) Adding all we get u1 + u2 + u3 + … + un = f(n+1) - f(1) because the remaining terms cancel each other. Note: This method can also be used in finding out the sum of an infinite series, if the series is convergent. Thus if Sn be the sum to n terms of a given series, then the sum to infinity will be as follows lim lim S f (n 1) f (1). n n n Sum of sines of a Series of n Angles which are in Arithmetic Progression: n sin n 1 sin sin( ) sin( 2 ) ... sin{ (n 1)} 2 sin sin 2 2 Sum of cosines of a Series of n Angles which are in Arithmetic Progression: n sin n 1 cos cos( ) cos( 2 ) ... cos{ (n 1)} 2 cos 2 sin 2 C + iS Method: The sum of many of the trigonometric series can be found out by a general method which is called C + iS method. Let us suppose that we need to find out the sum of any one of the following series, whether finite or infinite: C a0 cos a1 cos( ) a2 cos( 2 ) ... and S a0 sin a1 sin( ) a2 sin( 2 ) ... The sine series is denoted by the first alphabet S and the cosine series is denoted by its first alphabet C. We multiply the sine series by i (iota) and then add it to the cosine series, we get C iS a0 cos a1 cos( ) a2 cos( 2 ) ... a0 sin a1 sin( ) a2 sin( 2 ) ... a0 (cos sin ) a1(cos( ) sin( )) a2 (cos( 2 ) sin( 2 )) ... i i( ) i( 2 ) a0e a1e a2e ... The last series is then transformed into one of the following series: 1. Arithmetic Series 2. Binomial Series 3. Exponential Series 4. Logarithmic Series 5. Sine or Cosine Series 6. Gregory Series Thus we express the sum of the given series in the form of A + iB and then equating the real and imaginary parts, we get the values of C and S. Previous Asked Questions in University Examinations 1. Find the sum of the sines of a series of angles which are in arithmetic progression. [LNMU 2019 (H)] 1 x 1 x 1 x 2. Sum the series tan x tan tan ... tan and find its limiting value when n 2 2 22 22 2n1 2n1 tends to infinity. [LNMU 2018(H)] 1 1 1 3. Sum the series cos cos(2 ) cos(3 ) cos(4 ) ...ad inf . [LNMU 2 3 4 2019(H)] .