Limits of Sequences ③ Infinite Series

The result of continuing to add the terms of a sequence indefinitely is called an infinite series. Although it appears to mean “adding infinitely many numbers,” it is precisely defined as the limit of the partial sums.

Definition of an Infinite Series

For a sequence {a_n}, let the sum through the nth term (the partial sum) be S_n = a_1 + a_2 + … + a_n.
　Infinite series Σ_{n=1}^∞ a_n = lim_{n→∞} S_n

Infinite Geometric Series

An infinite series formed from a geometric sequence with first term a and common ratio r is called an infinite geometric series.

• When |r| < 1: It converges, and its sum is a/(1 − r)
• When |r| ≧ 1 (and a ≠ 0): It diverges

Necessary Condition for a Convergent Series

If the series Σ a_n converges, then lim_{n→∞} a_n = 0. The converse is not true; the harmonic series Σ 1/n diverges.

Series Using Partial Fraction Decomposition

When 1/(n(n+1)) = 1/n − 1/(n+1), intermediate terms cancel each other (telescoping).
S_N = 1 − 1/(N+1) → 1 (N→∞)

Practice Problems

1. Find the sum of the infinite geometric series 2 + 4/3 + 8/9 + ….
2. When the infinite geometric series a/(1−r) has sum 6 and common ratio r=1/3, find a.
3. Find Σ_{n=1}^∞ 1/((2n−1)(2n+1)).