1. ## Series convergence/divergence

Hello,

I am supposed to show why the following series are convergent or divergent, but I didn't get very far with either direct comparison, root, or ratio tests:

1.
$\sum\limits_{n = 0}^\infty {\frac{{1}}
{{{n^2} + a^2 }}}$

I would have thought that by direct comparison I could show that (1) converges, but not sure. (2) leaves me at a complete loss, though.
2.
$\sum\limits_{n = 0}^\infty {n \left(\frac{{{2n+1} }}
{{{3n+1}}}\right)^n}$

any help appreciated.

thank you -
Oz.

2. you should use a direct comparison test for the first one, compare it with a p series... can you see which one? When you add something to the denominator the fraction gets smaller

3. Originally Posted by ozfingwe
Hello,

I am supposed to show why the following series are convergent or divergent, but I didn't get very far with either direct comparison, root, or ratio tests:

1.
$\sum\limits_{n = 0}^\infty {\frac{{1}}
{{{n^2} + a^2 }}}$

I would have thought that by direct comparison I could show that (1) converges, but not sure. (2) leaves me at a complete loss, though.
2.
$\sum\limits_{n = 0}^\infty {n \left(\frac{{{2n+1} }}
{{{3n+1}}}\right)^n}$

any help appreciated.

thank you -
Oz.
1.
$n^2+a^2 \geq n^2$ for every $a \in R$
$\frac{1}{n^2+a^2} \leq \frac{1}{n^2}$
Now, You are supposed to complete it.

2.
After applying tha Root Test, you will face the following limit:
$\frac{2}{3}\lim_{n\to\infty} n^{\frac{1}{n}}$.
Evaluate this limit, and hance, determine the convergence of the series by using the concept of the Root Test.

**Edit** :
Ohhh sorry I did not read the whole question, Try to use the integral test to the first one.
What is the tests which you are fimilar with ?

4. Originally Posted by General
1.
$n^2+a^2 \leq n^2$ for every $a \in R$
$\frac{1}{n^2+a^2} \geq \frac{1}{n^2}$
Now, You are supposed to complete it.

2.
After applying tha Root Test, you will face the following limit:
$\frac{2}{2}\lim_{n\to\infinity} n^{\frac{1}{n}}$.
Evaluate this limit, and hance, determine the convergence of the series by using the concept of the Root Test.

you swapped your $\leq$ and $\geq$ in the first

proof: let a=1, $n^2+1\not\leq n^2$

5. Originally Posted by artvandalay11
you swapped your $\leq$ and $\geq$ in the first

proof: let a=1, $n^2+1\not\leq n^2$
Thanks.
Acuatlly I mixed the \geq and \leq =)
Its Latex Typo =D

It has been edited.