Series A series is the sum of the terms of a sequence.

A partial sum, S n, adds only the first n terms. S 1 = a 1 S 2 = a 1 + a 2 S 3 = a 1 + a 2 + a 3 S n = a 1 + a 2 + a 3 + … + a n These partial sums form a sequence.  If the sequence of partial sums converges to a value, that value is the sum of the infinite series

Ex. Find using partial sums.

The last example is called a geometric series. Thm. Consider the geometric series If |r| ≥ 1, then the series diverges. If |r| < 1, then the series converges to

Ex.

Ex. Determine if converges using partial sums.  This is called a telescoping series.

Thm. nth Term Test If, then diverges. Ex.

Pract. 1. Is convergent or divergent? If convergent, find the sum. 2. Write as a rational number. 3. Show that diverges.