# Determine whether the series converges or diverges.

• Nov 26th 2012, 12:18 PM
moonman
Determine whether the series converges or diverges.
$\sum_{n=1}^{\infty}\frac{\sqrt{n^4 + 5n + 10}}{(3n+2)^2(5n+1)}$

Will use comparison test:

$\sum a_{n} = \sum_{n=1}^{\infty}\frac{\sqrt{n^4 + 5n + 10}}{(3n+2)^2(5n+1)}$

$\sum b_{n} = \frac{n^2}{30n^3}$

Is this correct series for $\sum b_{n}$ ?
• Nov 26th 2012, 12:24 PM
coolge
Re: Determine whether the series converges or diverges.
• Nov 26th 2012, 12:28 PM
moonman
Re: Determine whether the series converges or diverges.
Thanks, I had to change the latex
• Nov 26th 2012, 12:39 PM
Plato
Re: Determine whether the series converges or diverges.
Quote:

Originally Posted by moonman
$\sum_{n=1}^{\infty}\frac{\sqrt{n^4 + 5n + 10}}{(3n+2)^2(5n+1)}$

Will use comparison test:

$\sum a_{n} = \sum_{n=1}^{\infty}\frac{\sqrt{n^4 + 5n + 10}}{(3n+2)^2(5n+1)}$

$\sum b_{n} = \frac{n^2}{30n^3}$

Is this correct series for $\sum b_{n}$ ?

Just note that $\frac{\sqrt{n^4 + 5n + 10}}{(3n+2)^2(5n+1)}\le \frac{4n^2}{(3n+2)^2(5n+1)}$
• Nov 26th 2012, 12:41 PM
moonman
Re: Determine whether the series converges or diverges.
Quote:

Originally Posted by Plato
Just note that $\frac{\sqrt{n^4 + 5n + 10}}{(3n+2)^2(5n+1)}\le \frac{4n^2}{(3n+2)^2(5n+1)}$

How did you get the 4 in the numerator?
• Nov 26th 2012, 01:10 PM
Plato
Re: Determine whether the series converges or diverges.
Quote:

Originally Posted by moonman
How did you get the 4 in the numerator?

$\sqrt{n^4 + 5n + 10}\le\sqrt{n^4 + 5n^4 + 10n^4}=4n^2$
• Nov 26th 2012, 01:25 PM
moonman
Re: Determine whether the series converges or diverges.
Quote:

Originally Posted by Plato
$\sqrt{n^4 + 5n + 10}\le\sqrt{n^4 + 5n^4 + 10n^4}=4n^2$

How? I'm sorry I don't understand how you get 4n^2
• Nov 26th 2012, 01:32 PM
Plato
Re: Determine whether the series converges or diverges.
Quote:

Originally Posted by moonman
How? I'm sorry I don't understand how you get 4n^2

$4n^4+5n^4+10n^4=16n^4$

What is the square root of that?
• Nov 26th 2012, 01:41 PM
moonman
Re: Determine whether the series converges or diverges.
Oh gosh! Thanks, I feel like an idiot now. I did not follow the order of operations.