1. ## Tools in Analysis

I want to make a thread about all different tools and tricks that can help solve a difficult problem: finding limits, infinite sums, improper integral, ....

I want to begin with something that I did not learn so long ago but I liked it very much.

Given a power series $\sum_{n=0}^{\infty}a_nx^n$ most people compute $\lim \left| \frac{a_{n+1}x^{n+1}}{a_nx^n} \right|$ to determine its radius of convergence.

But here is a stronger method (it is really stated in something called the limit superior, but since not everyone knows it I will state a weaker version for more people to befinit).

The above was called the "ratio test" the following tool is called the "root test". Instead we compute $\lim \left| a_n x^n \right|^{1/n}$. And the same conditions apply, i.e. if $<1$ then convergence and if $>1$ then divergence.

However, we just need to know one very important limit that should be memorized $\lim n^{1/n} = 1$.

So for example given,
$\sum_{n=0}^{\infty} \frac{2^nx^n}{n^n}$.
Doing this by the ratio test is messy, instead use the root test.
$\lim \left| \frac{2^nx^n}{n^n} \right|^{1/n} = \lim \left| \frac{2x}{n} \right| =0$ which means $R=+\infty$.

The reason why it is stronger than the ratio test is because in that test we assume the coefficients in the power series are not zero. But here that does not matter.

2. Just a suggestion.... Why don't you start this as a Wiki page?

-Dan

3. Originally Posted by topsquark
Just a suggestion.... Why don't you start this as a Wiki page?

-Dan
I do not really know how to use MHFWiki

4. Originally Posted by ThePerfectHacker
I do not really know how to use MHFWiki
It's pretty much the same as posting here. I went to Wikipedia to get a few other tricks to make the page look nicer.

-Dan

5. ## Stolz-Cesaro Theorem (Lemma)

6. Red_dog know any more tricks to use?

Here is another cool test. (Dirichlet's Test).

Using this test you can prove the convergence of,
$\sum_{n=1}^{\infty} \frac{\sin n}{n}$.
(Try doing it without this test. This is supposed to be a really hard converge/diverge problem*).

*)Yet amazingly, it is possible to find its exact sum (something involving $\ln(\sin 1)$). I seem somebody on AoPs do it using some heavy tools (Fourier Analysis).