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Types of Series: p-Series, Harmonic Series, Geometric Series, Decimal Expansion for AP Calculus

based on 4 ratings
By — McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for these concepts can be found at: Series Practice Problems for AP Calculus

Types of Series Main Concepts

p-Series, Harmonic Series, Geometric Series, Decimal Expansion

p-Series

The p-series is a series of the form . The p-series converges when p > 1, and diverges when 0 < p ≤ 1.

Harmonic Series

The harmonic series is a p-series with p = 1.

The harmonic series diverges.

Geometric Series

A geometric series is a series of the form where a ≠ 0. A geometric series converges when |r| < 1. The sum of the first n terms of a geometric series is . The sum of the series .

Example 1

Determine whether the series converges. is a geometric series with a = 1 and r = . Since r > 1, the series diverges.

Example 2

Find the tenth partial sum of the series

While it is possible to extend the terms of the series and directly compute the tenth partial sum, it is quicker to recognize that this is a geometric series. The ratio of any two subsequent terms is r = and the first term is a = 12.

Example 3

Find the sum of the series Since is a geometric series with a = 12 and .

Decimal Expansion

The rational number equal to the repeating decimal is the sum of the geometric series that represents the repeating decimal.

Example

Find the rational number equivalent to .

Step 1:  

Step 2:   is a geometric series with a = and r = The sum of the series is .

Step 3:  

Practice problems for these concepts can be found at: Series Practice Problems for AP Calculus

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