# Which tests to use for convergence?

• May 18th 2010, 04:57 AM
simpleas123
Which tests to use for convergence?
Hey all i was wondering what tests to use for these series, as some of them i have no idea how to tackle, hopefully you can help me out :).

http://img33.imageshack.us/img33/7615/67912800.jpg

Pointing me in the right direction will help me greatly for my exam tomorrow!
Thanks people.

I had no idea how to attempt any of them, and my lecture notes aint helping me too. For the sin ones, would i use a comparison with 1/n? because its sin (n) ?
• May 18th 2010, 05:28 AM
tonio
Quote:

Originally Posted by simpleas123
Hey all i was wondering what tests to use for these series, as some of them i have no idea how to tackle, hopefully you can help me out :).

http://img33.imageshack.us/img33/7615/67912800.jpg

Pointing me in the right direction will help me greatly for my exam tomorrow!
Thanks people.

I had no idea how to attempt any of them, and my lecture notes aint helping me too. For the sin ones, would i use a comparison with 1/n? because its sin (n) ?

1) $\displaystyle \left|\frac{n-\sin^2n}{n^3}\right|\leq \frac{n+1}{n^3}\leq 2\frac{1}{n^2}$ ...comparison test for absolute convergence

2) $\displaystyle \frac{1}{2^n+1}\leq \left(\frac{1}{2}\right)^n$ ...comparison test

3) $\displaystyle \lim_{n\to\infty}\frac{n}{\ln n} \neq 0$ ... in fact, $\displaystyle \frac{n}{\ln n}\xrightarrow [n\to\infty]{}\infty$

4) $\displaystyle \frac{n+1/n}{n!+n^3}\leq \frac{2n}{n!}=2\frac{1}{(n-1)!}$ , and then use the ratio test to show this last series converges.

Tonio
• May 18th 2010, 05:44 AM
simpleas123
Quote:

Originally Posted by tonio
1) $\displaystyle \left|\frac{n-\sin^2n}{n^3}\right|\leq \frac{n+1}{n^3}\leq 2\frac{1}{n^2}$ ...comparison test for absolute convergence

2) $\displaystyle \frac{1}{2^n+1}\leq \left(\frac{1}{2}\right)^n$ ...comparison test

3) $\displaystyle \lim_{n\to\infty}\frac{n}{\ln n} \neq 0$ ... in fact, $\displaystyle \frac{n}{\ln n}\xrightarrow [n\to\infty]{}\infty$

4) $\displaystyle \frac{n+1/n}{n!+n^3}\leq \frac{2n}{n!}=2\frac{1}{(n-1)!}$ , and then use the ratio test to show this last series converges.

Tonio

Thank you very much! +rep'd