Math 162

Class 20

Geometric Series

\[\sum_{i=0}^\infty ar^i\]

Converges to \(\displaystyle\frac{a}{1-r}\) if \(\left\lvert r\right\rvert < 1\).

Diverges if \(\left\lvert r\right\rvert \ge 1\)

Telescoping Series

\[\sum_{i=k}^\infty \left(b_i - b_{i+1}\right)\]

\(s_n = b_k - b_{n+1}\)

Converges to \(b_k - \displaystyle\lim_{n\to\infty} b_n\) if this limit exists.

A discrete version of the FTC!

Example

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

Harmonic Series

\[\sum_{i = 1}^\infty \frac{1}{i}\]

divergent

\(\displaystyle s_{2^n} > \frac{n}{2}\)

The \(n\)-th term test for divergence

If the \[\lim_{n\to\infty} a_n\] is not 0, then the series \[\sum_{i=k}^\infty a_i\] diverges.

Dropping terms

Let \(N > k\) are two integers. Then \(\displaystyle \sum_{i=k}^\infty a_i\) converges if and only if \(\displaystyle\sum_{i=N}^\infty a_i\) converges.

Proof: Let \(s_n\), \(n = k, k+1, \dots\) be the sequence of partial sums of the series \(\displaystyle \sum_{i=k}^\infty a_i\), and let \(r_n\), \(n=N, N+1, \dots\) be the sequence of partial sums of the series \(\displaystyle \sum_{i=N}^\infty a_i\).

Then \(s_n = r_n + s_{N-1}\) for every \(n \ge N\).

Series with non-negative terms

If \(a_i \ge 0\) for every \(i = k, k+1, \dots\), we can visualize the series as an area:

k k +1 k +2 k +3 k +4 k +5 k +6 k +7 k +8 k +9 k +10 k +11 a k a k +1 a k +2 a k +3 a k +4 a k +5 a k +6 a k +7 a k +8 a k +9 a k +10 a k +11

Comparison test

If \(0 \le a_n \le b_n\) for each \(n \ge k\) and if \(\displaystyle\sum_{i=k}^\infty b_i\) converges, then \(\displaystyle\sum_{i=k}^\infty a_i\) also converges.

If \(a_n \ge b_n \ge 0\) for each \(n \ge k\) and if \(\displaystyle\sum_{i=k}^\infty b_i\) diverges, then \(\displaystyle\sum_{i=k}^\infty a_i\) also diverges.

Example

\(\displaystyle\sum_{i=0}^\infty \frac{1}{3^i + 5}\)

Example

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

Comparing to other areas

Suppose \(a_n = f(n)\) for \(n = k, k+1, \dots\), where \(f\) is a decreasing and continuous function.

k k +1 k +2 k +3 k +4 k +5 k +6 k +7 k +8 k +9 k +10 k +11 a k a k +1 a k +2 a k +3 a k +4 a k +5 a k +6 a k +7 a k +8 a k +9 a k +10 a k +11 ∞ k f ( x ) dx ≤ ∞ i = k a i ∞ k f ( x ) dx ≤ ∞ i = k a i ≤ ∞ k − 1 f ( x ) dx f ( x )

Integral Test

Suppose \(a_n = f(n)\) for \(n = k, k+1, \dots\), where \(f\) is a decreasing and continuous function.

  • If \(\displaystyle\int_{k-1}^\infty f(x)\;dx\) converges, then so does \(\displaystyle \sum_{i=k}^\infty a_i\).

  • If \(\displaystyle\int_{k}^\infty f(x)\;dx\) diverges, then so does \(\displaystyle \sum_{i=k}^\infty a_i\).

Example

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

Example

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

Example (\(p\)-series)

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

Example

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

Example

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

Example

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

Example

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

Limit Comparison Test

Suppose \(a_n \ge 0\) and \(b_n \ge 0\) for all \(n \ge k\).

  • If \(\displaystyle\lim_{n\to\infty}\frac{a_n}{b_n} = L > 0\) then the series \(\displaystyle\sum_{i=k}^\infty a_i\) and \(\displaystyle\sum_{i=k}^\infty b_i\) either both converge or both diverge.
  • If \(\displaystyle\lim_{n\to\infty}\frac{a_n}{b_n} = 0\) and the series \(\displaystyle\sum_{i=k}^\infty b_i\) converges, then \(\displaystyle\sum_{i=k}^\infty a_i\) also converges.
  • If \(\displaystyle\lim_{n\to\infty}\frac{a_n}{b_n} = \infty\) and the series \(\displaystyle\sum_{i=k}^\infty b_i\) diverges, then \(\displaystyle\sum_{i=k}^\infty a_i\) also diverges.

Example

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

Example

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

Example

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