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# Warm Up - PowerPoint PPT Presentation

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.

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### 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?

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 S1, S2, S3, S4, S5, …and an expression for Sn

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 an 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

1.

2.

3.

Find the sum of the series series.

This is called a telescoping series.

Find the sum of the series series.

This is another telescoping series.

• 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: series.

When asked if a series converges or diverges:

• Do the nth term test for divergence.

• If the series if geometric, find r and determine whether the series converges or diverges. If converges, find the sum.

• 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.

Test series.

Series

Condition(s) for convergence

Conditions for divergence