Alternating Series

The Essentials

A series that switches between negative and positive as n increases is called an alternating series. The alternating series can be tested to see if it is convergent or divergent. If you have an alternating series following the two patterns shown below.

\[ \sum_{n=1}^{\infty}(-1)^{n+1}b_n \]

or

\[ \sum_{n=1}^{\infty}(-1)^nb_n \]

Then you can know that your series converges if it follows both of the following conditions. If it doesn't follow both conditions, the test is considered inconclusive.

  1. \( 0 < b_{n+1}\leq b_n \) for all \( n\geq 1 \)
  2. \( \lim_{n\to \infty}b_n=0 \)

Example

If we have an alternating series denoted as we can confirm the series is convergent or if the test is inconclusive by comparing to the two requirement of convergence.

\[ \sum_{n=1}^{\infty}(-1)^n(\frac{1}{n^3}) \]

We can see that the first statement is true because
\( 0 < \frac{1}{(n+1)^3}\leq \frac{1}{n^3} \)

The second statement is also true because
\( \lim_{n\to \infty}\frac{1}{n^3}=0\)

Therefore, the series \( \sum_{n=1}^{\infty}(-1)^n(\frac{1}{n^3}) \) is convergent.

Practice

Evaluate these expressions:

  1. Determine the result of the alternating series test for \( \sum_{n=1}^{\infty}(-1)^n(\frac{1}{n}) \).
  2. Determine the result of the alternating series test for \( \sum_{n=1}^{\infty}(-1)^n(\frac{3n+2}{n-5}) \)

Solutions:

  1. Convergent
  2. Inconclusive

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