Determine whether the series converges or diverges.

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

Will use comparison test:

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

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

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

Re: Determine whether the series converges or diverges.

Re: Determine whether the series converges or diverges.

Thanks, I had to change the latex

Re: Determine whether the series converges or diverges.

Quote:

Originally Posted by

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

Will use comparison test:

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

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

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

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

Re: Determine whether the series converges or diverges.

Quote:

Originally Posted by

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

How did you get the 4 in the numerator?

Re: Determine whether the series converges or diverges.

Quote:

Originally Posted by

**moonman** How did you get the 4 in the numerator?

$\displaystyle \sqrt{n^4 + 5n + 10}\le\sqrt{n^4 + 5n^4 + 10n^4}=4n^2$

Re: Determine whether the series converges or diverges.

Quote:

Originally Posted by

**Plato** $\displaystyle \sqrt{n^4 + 5n + 10}\le\sqrt{n^4 + 5n^4 + 10n^4}=4n^2$

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

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

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

What is the square root of that?

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.