October 16, 2016

## Idea of the ratio test

What's the defining characteristic of a geometric series? A series $$\displaystyle \sum_{n=0}^{\infty} a_n$$ is geometric if ______________

If the ratio of successive terms of a positive series eventually approaches a constant $$L<1$$, then the series can be bounded above by a convergent geometric series with ratio $$0<r<1$$.

## Ratio test

Theorem: Let $$\lbrace a_n \rbrace$$ be a positive sequence where $$\displaystyle\lim_{n\to \infty} \frac{a_{n+1}}{a_n} = L$$.

1. If $$L<1$$, then the series $$\displaystyle\sum_{n=1}^{\infty} a_n$$ converges.

2. If $$L>1$$, then the series $$\displaystyle\sum_{n=1}^{\infty} a_n$$ diverges.

3. If $$L=1$$, then the ratio test is inconclusive, so try another method.

Informal summary: If a series is eventually positive and eventually like a convergent geometric series, it converges.

## Ratio test proof idea (convergent case)

Choose a fixed $$r$$ so that $$L < r < 1$$. Then $$\displaystyle \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = L$$ implies there is some $$k$$ so that $$a_{n+1} < r a_n$$ for all $$n > k$$. Thus, $\begin{array}{l} a_{k+1} \leq r a_k \\ a_{k+2} \leq r a_{k+1} \leq r^2 a_k \\ a_{k+3} \leq r a_{k+2} \leq r^2 a_{k+1} \leq r^3 a_k \end{array}$ and hence the tail $a_k + a_{k+1} + a_{k+2} + a_{k+3} + \cdots$ is bounded above by a convergent geometric series $a_k + r a_k + r^2 a_k + r^3 a_k + \cdots.$

## Examples

Determine whether these series converge. Don't forget to check validity conditions!

1. $$\displaystyle \sum_{n=1}^{\infty} \frac{n! n^2}{(2n)!}$$

2. $$\displaystyle \sum_{n=1}^{\infty} \frac{5^n-3n}{4^n}$$

## Idea of the root test

In a geometric series, $$a_n = r^n = r \cdot r \cdot r \cdots r$$ (with $$n$$ factors), and thus $$\sqrt[n]{a_n} = (r^n)^{1/n} = r$$.

In an arbitrary series with all positive terms, split each term, $$a_n$$, of the series into $$n$$ equal size factors. If each factor $$\sqrt[n]{a_n}$$ is less than a fixed positive ratio $$r<1$$, then the series is bounded above by a convergent geometric series.

## Root test

Theorem: Let $$\lbrace a_n \rbrace$$ be a positive sequence where $$\displaystyle\lim_{n\to \infty} \sqrt[n]{a_n} = L$$.

1. If $$L<1$$, then the series $$\displaystyle\sum_{n=1}^{\infty} a_n$$ converges.

2. If $$L>1$$, then the series $$\displaystyle\sum_{n=1}^{\infty} a_n$$ diverges.

3. If $$L=1$$, then the ratio test is inconclusive, so try another method.

Informal summary: If a series is eventually positive and eventually like a convergent geometric series, it converges.

## Root test proof idea (convergent case)

Choose a fixed $$r$$ so that $$L < r < 1$$. Then $$\displaystyle \lim_{n \to \infty} \sqrt[n]{a_n} = L$$ implies there is some $$k$$ so that $$\sqrt[n]{a_n} < r$$ for all $$n > k$$. Thus, $$a_n < r^n$$ for all $$n > k$$ and hence the tail $a_k + a_{k+1} + a_{k+2} + a_{k+3} + \cdots$ is bounded above by a convergent geometric series $r^k + r^{k+1} + r^{k+2} + r^{k+3} + \cdots.$

## Roots and relative growth rates

$$\displaystyle \lim_{n \to \infty} \sqrt[n]{C} = 1$$ for any constant $$C$$

$$\displaystyle \lim_{n \to \infty} \sqrt[n]{n^p} = 1$$ for any $$p>0$$

$$\displaystyle \lim_{n \to \infty} \sqrt[n]{b^n} = b$$ for any constant $$b$$

$$\displaystyle \lim_{n \to \infty} \sqrt[n]{n!} = \infty$$

$$\displaystyle \lim_{n \to \infty} \sqrt[n]{n^n} = \infty$$

## Examples

Determine whether these series converge. Don't forget to check validity conditions!

1. $$\displaystyle \sum_{n=1}^{\infty} \left( \frac{n+2}{2n+1} \right)^n$$

2. $$\displaystyle \sum_{n=0}^{\infty} \frac{5^n (3n^2+1)}{2^n n!}$$

3. $$\displaystyle \sum_{n=2}^{\infty} \frac{n^2}{(\ln n)^n }$$