# Thread: Prove this sum tends to zero

1. ## Prove this sum tends to zero

Show $\sum_{i=2}^{n} \frac{i}{n^2 \log i} \rightarrow 0$ as $n \rightarrow \infty$.

I can solve a problem if I can prove this sum tends to zero as n tends to infinity but I'm struggling with the last stage. I'm not sure if I'm missing something obvious...

2. Originally Posted by Boysilver
Show $\sum_{i=2}^{n} \frac{i}{n^2 \log i} \rightarrow 0$ as $n \rightarrow \infty$.

I can solve a problem if I can prove this sum tends to zero as n tends to infinity but I'm struggling with the last stage. I'm not sure if I'm missing something obvious...
Can you evaluate

$\sum_{i = 2}^{\infty} \frac{i}{n^2 \log{i}}$?

3. Originally Posted by Prove It
Can you evaluate

$\sum_{i = 2}^{\infty} \frac{i}{n^2 \log{i}}$?

I can't see how that helps, not to mention that that infinite series diverges so he won't be able to evaluate it, nor anyone else will.

Tonio

4. How about writing it as:

$\lim_{n\to\infty}\left\{\frac{\displaystyle{\sum_{ k=2}^n \frac{k}{\log(k)}}}{n^2}\right\}$

The general expression for the sum is $\frac{2}{\log(2)}+\frac{3}{\log(3)}+\cdots+\frac{n }{\log(n)}$

Isn't each of those less than $\frac{n}{\log(n)}$ and you have less than $n$ of them then what does that say about the limit?