Learn Series

To learn about the terms of a sequence connected by the sign plus(+) is called a series. Thus a1 + a2 + a3 + … an + … is an infinite series. Usually the symbol Σ an is used to denote a series.

                                   An example for series is zeno's dichotomy. Let us learn about Arithmetic Progression(A.P) and Geometric Progression(G.P) series.

 

Learn sum to n terms of an A.P series:

 

Let Sn denote the sum of the terms of the A.P a, a + d, a + 2d, …. a+(n–1) d


                        Sn = a + (a + d) + (a + 2d)  + …. .+  [a + (n–1)d]  --------------- (a)


Writing this series in the reverse order


                         Sn = [a + (n – 1)d] + [a + (n – 2)d] + …. + a    --------------------(b)


Adding (a) and (b), we get


                             2Sn = [2a + (n – 1)d] + [2a + (n – 1)d] + …. + [2a + (n – 1)d]


                                   = n [2a + (n – 1) d]

                              Sn = n/2 [2a + (n-1)d]
      
                              Sn = n/2 [a + a + (n - 1)d]

                              Sn = n/2[a + l]

where l is tn = a + (n – 1) d = last term

 

 

 

Learn sum to n terms of a G.P series:

 

Let Sn denote the sum of n terms of the G.P a, ar, ar2, ….

                        Sn = a + ar + ar2 + …. + arn-1 -------------------------(a)

Multiplying both sides by r

                         r. Sn = ar + ar2 + ar3 + … + arn --------------------(b)

(b) – (a) ⇒ r. Sn – Sn = arn – a, Sn (r – 1) = a(rn – 1)

                               If r > 1, Sn = a(rn - 1) / r - 1

                               If r < 1,  Sn = a(1 - rn ) / r - 1

                               If r = 1, Sn = na
If r < 1 say r = 1/2, then r2 = 1/4, r3 = 1/8, …. rn = 1/2n is very small and rn → 0 when n is very large or

r < 1 ⇒ rn  → 0  as n → ∞

So, therefore sum of infinite geometric series = S = a / (1 - r).