W08 - Examples

Simple divergence test examples

Simple divergence test: examples

(a) Consider: $\sum _{n=1}^{\infty }\frac{n}{4n+1}$

This diverges by the SDT because $a_n\to \frac{1}{4}$ and not $0$.

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(b) Consider: $\sum _{n=1}^{\infty }(-1{)}^{n-1}\frac{n}{n+1}$

This diverges by the SDT because ${\lim }_{n\to \infty }{a}_n=\text{DNE}$.

We can say the terms “converge to the pattern $+1,-1,+1,-1,\dots$,” but that is not a limit value.

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p-series examples

p-series examples

By finding $p$ and applying the $p$-series convergence properties:

We see that ${\sum }_{n=1}^{\infty }\frac{1}{n^{1.1}}$ converges: $p=1.1$ so $p>1$

But ${\sum }_{n=1}^{\infty }\frac{1}{\sqrt{n}}$ diverges: $p=1/2$ so $p\le 1$

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Integral test - pushing the envelope of convergence

Integral test - pushing the envelope of convergence

Does $\sum _{n=2}^{\infty }\frac{1}{n\ln n}$ converge?

Does $\sum _{n=2}^{\infty }\frac{1}{n(\ln n{)}^2}$ converge?

Notice that $\ln n$ grows very slowly with $n$, so $\frac{1}{n\ln n}$ is just a little smaller than $\frac{1}{n}$ for large $n$, and similarly $\frac{1}{n(\ln n{)}^2}$ is just a little smaller still.

Solution

(1) The two series lead to the two functions $f(x)=\frac{1}{x\ln x}$ and $g(x)=\frac{1}{x(\ln x{)}^2}$.

Check applicability.

Clearly $f(x)$ and $g(x)$ are both continuous, positive, decreasing functions on $x\in [2,\infty )$.

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(2) Apply the integral test to $f(x)$.

Integrate $f(x)$:

$$\begin{array}{rl}{\int }_2^{\infty }\frac{1}{x\ln x}dx& \gg \gg {\int }_{u=\ln 2}^{\infty }\frac{1}{u}du\\ & \\ & \gg \gg \underset{R\to \infty }{\lim }\ln u{|}_{\ln 1}^R\gg \gg \infty \end{array}$$

Conclude: ${\sum }_{n=2}^{\infty }\frac{1}{n\ln n}$ diverges.

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(3) Apply the integral test to $g(x)$.

Integrate $g(x)$:

$$\begin{array}{rl}{\int }_2^{\infty }\frac{1}{x(\ln x{)}^2}dx& \gg \gg {\int }_{u=\ln 2}^{\infty }\frac{1}{u^2}du\\ & \\ & \gg \gg \underset{R\to \infty }{\lim }-u^{-1}{|}_{\ln 2}^R\gg \gg \frac{1}{\ln 2}\end{array}$$

Conclude: ${\sum }_{n=2}^{\infty }\frac{1}{n(\ln n{)}^2}$ converges.

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