Warm Up Write the explicit formula for the series. Evaluate.

Introduction to Series What is a series? What does it mean for a series to converge? What are geometric and telescoping series? What is the nth term test?

Infinite Series

Partial Sums

A series can either converge or diverge. If the sequence of nth partial sums converges to A, then the series converges. The limit of S is called the sum of the series. If the sequence of nth partial sums diverges, then the series diverges.

Ex: Find S 1, S 2, S 3, S 4, S 5, …and an expression for S n Therefore, the series converges to 1. note: If the limit of the sequence is NOT 0, then the sum of the series must diverge.

This example was a geometric series because a n is an exponential function. A geometric series with ratio (base) r diverges if |r| > 1 Converges if 0 < r < 1 If it converges then the series converges to

Converge/Diverge? If converge, tell the limit of the series

Find the sum of the series This is called a telescoping series.

Find the sum of the series This is another telescoping series.

Let’s revisit the “note” we talked about before. If the limit of the sequence is not 0, then the series diverges. The contrapositive of this statement must also be true: If the series converges, then the limit of the sequence is 0. However, the converse (and inverse) do not have to be true…and are NOT in this case. Just because the limit of the sequence is 0, the series can still diverge. All of this information is classified as the “nth term test for divergence.” Always use this as your first step in answering the converge/diverge question for series.

In Summary: When asked if a series converges or diverges: 1.Do the nth term test for divergence. 2.If the series if geometric, find r and determine whether the series converges or diverges. If converges, find the sum. 3.If the series is a rational function and the denominator can be factored, separate the ratio by partial fractions and determine if it is a telescoping series. If it is, find the value to which it converges.

TestSeries Condition(s) for convergence Conditions for divergence Example & Comments