Math 162

Class 21

“Known” series

• Known sums:
  • Geometric series
  • Telescoping series
• Known behavior:
  • \(p\)-series

Tests

• The \(n\)-th term test for divergence
• Series with non-negative terms:
  • Integral test
  • Comparison test
  • Limit comparison test

Alternating series

• \(b_n > 0\) for \(n = k, k+1, \dots\)
• \(a_n = (-1)^n b_n\) for \(n = k, k+1, \dots\) or \(a_n = (-1)^{n+1} b_n\) for \(n = k, k+1, \dots\)
• \(\displaystyle \sum_{i=k}^\infty a_n\) is an alternating series.

Alternating Series Test

Let \(\displaystyle\sum_{i=k}^\infty a_i\) be an alternating series such that

• \(a_n = (-1)^n b_n\) for \(n = k, k_1, \dots\) or \(a_n = (-1)^{n+1} b_n\) for \(n = k, k_1, \dots\)
• \(b_n = \left\lvert a_n\right\rvert\) is decreasing
• \(\displaystyle \lim_{n\to\infty} b_n = 0\)

Then \(\displaystyle\sum_{i=k}^\infty a_i\) is convergent.

Alternating Harmonic Series

\(\displaystyle\sum_{i=1}^\infty (-1)^i\frac{1}{i}\)

Example

\(\displaystyle\sum_{i=1}^\infty (-1)^i\frac{i}{i+1}\)

Example

\(\displaystyle\sum_{i=1}^\infty (-1)^i\frac{i}{i^2+1}\)

Example

\(\displaystyle\sum_{i=0}^\infty (-1)^i\frac{1}{\sqrt{i^2+1}}\)

Definition

Let \(\displaystyle\sum_{i=k}^\infty a_i\) be a series such that the series \(\displaystyle\sum_{i=k}^\infty \left\lvert a_i\right\rvert\) converges.

Then \(\displaystyle\sum_{i=k}^\infty a_i\) converges.

We say that \(\displaystyle\sum_{i=k}^\infty a_i\) converges absolutely, or that it is absolutely convergent.

Definition

Let \(\displaystyle\sum_{i=k}^\infty a_i\) be a convergent series such that the series \(\displaystyle\sum_{i=k}^\infty \left\lvert a_i\right\rvert\) diverges.

We say that \(\displaystyle\sum_{i=k}^\infty a_i\) converges conditionally, or that it is conditionally convergent.

Examples

What is it about?

Let \(\displaystyle\sum_{i=k}^\infty a_i\) be a convergent series with some positive and some negative terms.

• The series is absolutely convergent if and only if the sum of positive terms and the sum of negative terms are both convergent.
• The series is conditionally convergent if and only if the sum of positive terms is \(\infty\) and the sum of negative terms is \(-\infty\).

Rearrangements