# methods on determining convergence or divergence

• Jul 9th 2007, 03:52 PM
Possible actuary
methods on determining convergence or divergence
Can you use either taking the limit or solve the integral to find out if $\displaystyle \sum_{n=1}^\infty\frac{n}{\sqrt{n^2+1}}$ diverges or converges?
• Jul 9th 2007, 04:13 PM
Jhevon
Quote:

Originally Posted by Possible actuary
Can you use either taking the limit or solve the integral to find out if $\displaystyle \sum_{n=1}^\infty\frac{n}{\sqrt{n^2+1}}$ diverges or converges?

the limit test for divergence is the easiest method.

$\displaystyle \lim_{n \to \infty} \frac {n}{ \sqrt {n^2 + 1}} \neq 0$ thus the series diverges
• Jul 9th 2007, 06:23 PM
curvature
Quote:

Originally Posted by Possible actuary
Can you use either taking the limit or solve the integral to find out if $\displaystyle \sum_{n=1}^\infty\frac{n}{\sqrt{n^2+1}}$ diverges or converges?

The problem becomes harder if we consider $\displaystyle \sum_{n=1}^\infty\frac{n}{\sqrt{n^3+1}}$ or $\displaystyle \sum_{n=1}^\infty\frac{n}{\sqrt{n^4+1}}$