# Convergence and divergence of a series

1. Oct 10, 2014

### smart_worker

B]1. The problem statement, all variables and given/known data[/B]
Find whether the series is convergent or divergent

2. Relevant equations

3. The attempt at a solution

By ratio test I have,

I would apply L'Hôpital's rule to find the value of limit but before that how do i simplify the expression? It has fractional part both in the numerator as well as in the denominator.

Last edited: Oct 10, 2014
2. Oct 10, 2014

### Ray Vickson

Just use elementary high-school algebra. Alternatively, look more carefully at the problem before even starting. Maybe the ratio test won't work; there are times when it doesn't.

3. Oct 10, 2014

### RUber

Consider breaking it into two sums...the sum of convergent series is convergent, however if one diverges, the sum of the two diverges (generally).

4. Oct 10, 2014

If you want to simplify the large fraction in
$$\lim_{n\to \infty} \left(\dfrac{\left(\dfrac{(n+1)^2}{2^{n+1}} + \dfrac{1}{(n+1)^2}\right)}{\dfrac{n^2}{2^n} + \dfrac 1 {n^2}}\right)$$

treat it the way you would a complex fraction. As has been stated above, however, I'm not sure this approach will generate a positive result.

Think about the idea that if both $\sum_{i=1}^\infty a_n$ and $\sum_{i=1}^\infty b_n$ are absolutely convergent then
$\sum_{i=1}^\infty \left(a_n + b_n \right)$ is absolutely convergent.

5. Oct 10, 2014

### LCKurtz

Not sure what you mean by "generally" other than perhaps it means "sometimes" because you know it's false in general.