The ratio test for series

## The ratio test for series
            ---
            ## Ratio test
            #### Example
            $\sum\_{n=1}^\infty \frac{1}{2^n}$ converges.
            #### Proof
            The sequence of partial sums is $s_n = \frac12+\frac14+ \ldots+ \frac1{2^n}=\frac{1-(\frac12)^n}{1-\frac12} - 1 \to \frac1{1/2}-1=1$, as $n\to\infty$.   □
            
            ---
            ## Ratio test
            #### Example
            $\sum\_{n=1}^\infty \frac{1}{2^n}$ converges.
            #### Proof
            The sequence of partial sums is $s_n = \frac12+\frac14+ \ldots+ \frac1{2^n}=\frac{1-(\frac12)^n}{1-\frac12} - 1 \to \frac1{1/2}-1=1$, as $n\to\infty$.   □

Here the ratio between consecutive terms is $\frac12$ and $0<\frac12<1$.
            The same argument would work for convergence of $\sum\_{n=1}^\infty r^n$ with any $r\in(-1,1)$.
            
            ---
            ## Ratio test
            #### Example
            $\sum\_{n=1}^\infty \frac{1}{2^n}$ converges.
            #### Proof
            The sequence of partial sums is $s_n = \frac12+\frac14+ \ldots+ \frac1{2^n}=\frac{1-(\frac12)^n}{1-\frac12} - 1 \to \frac1{1/2}-1=1$, as $n\to\infty$.   □

Here the ratio between consecutive terms is $\frac12$ and $0<\frac12<1$.
            The same argument would work for convergence of $\sum\_{n=1}^\infty r^n$ with any $r\in(-1,1)$.

But we can do better.

#### Theorem
            The series $\sum\_{n=1}^\infty a_n$ with $\lim\_{n\to\infty} \left\lvert\frac{a\_{n+1}}{a_n}\right\rvert = \ell$ converges if $\ell<1$, but diverges if $\ell>1$.
            
            ---
            ## Ratio test
            #### Theorem
            The series $\sum\_{n=1}^\infty a_n$ with $\lim\_{n\to\infty} \left\lvert\frac{a\_{n+1}}{a_n}\right\rvert = \ell$ converges if $\ell<1$, but diverges if $\ell>1$.
            #### Examples
            $\sum\_{n=1}^\infty \frac1{3^n}$ converges, but $\sum\_{n=1}^\infty 2^n$ diverges.
            
            ---
            ## Ratio test
            #### Theorem
            The series $\sum\_{n=1}^\infty a_n$ with $\lim\_{n\to\infty} \left\lvert\frac{a\_{n+1}}{a_n}\right\rvert = \ell$ converges if $\ell<1$, but diverges if $\ell>1$.
            #### Examples
            $\sum\_{n=1}^\infty \frac1{3^n}$ converges, but $\sum\_{n=1}^\infty 2^n$ diverges.
            #### Examples
            Using the ratio test, and possibly some other tests, decide convergence of:

$\sum\_{n=1}^\infty \frac{2^n}{3^n},\\; \sum\_{n=1}^\infty \frac{1}{n\sqrt n},\\; \sum\_{n=1}^\infty n\^{100}(\frac9{10})^n,\\; \sum\_{n=1}^\infty \frac{n\^{100}}{(\frac9{10})^n},\\; \sum\_{n=1}^\infty \frac{2^n + n^5}{n\sin(n) + 3^n},\\; \sum\_{n=1}^\infty \frac{2^n + 4n}{5 - 2^n}$