Math 162

Class 22

Ratio Test

Suppose \(\displaystyle\sum_{i=k}^\infty a_i\) be a series with nonzero terms such that

\[\displaystyle \rho = \lim_{n\to\infty} \left\lvert \frac{a_{n+1}}{a_n}\right\rvert\]

exists or is infinite.

• If \(0 \le \rho < 1\), the series converges absolutely.
• If \(\rho > 1\) (including infinity), the series diverges.
• If \(\rho = 1\), there is no conclusion.

Root Test

Suppose \(\displaystyle\sum_{i=k}^\infty a_i\) be a series such that

\[\displaystyle \rho = \lim_{n\to\infty} \sqrt[n]{\left\lvert a_n\right\rvert}\]

exists or is infinite.

• If \(0 \le \rho < 1\), the series converges absolutely.
• If \(\rho > 1\) (including infinity), the series diverges.
• If \(\rho = 1\), there is no conclusion.

Example

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

Example

\(\displaystyle\sum_{i=1}^\infty \frac{(-1)^i}{i2^i}\)

Example

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

Example

\(\displaystyle\sum_{i=0}^\infty \frac{i^i}{i!}\)

Example

\(\displaystyle\sum_{i=0}^\infty \frac{3}{i^2 + 1}\)

Example

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

Challenge

\(\displaystyle\sum_{i=2}^\infty \frac{1}{i(\ln i)^3}\)

Challenge

\(\displaystyle\sum_{i=2}^\infty \frac{1}{i^3\ln i}\)

Challenge

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

Challenge

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