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$$\lim_{n\to\infty}\frac{1}{n}\sum_{k=2}^{n}\frac{1}{\log(k)}= 0$$

## A hint to a problem for convergence in probability: $\lim_{n \to \infty} \frac{1}{n} \sum_{k=2}^{n} \frac{1}{\log(k)} = 0$ - Question

https://math.stackexchange.com/questions/1977233
I would like a hint to show that $$\lim_{n \to \infty} \frac{1}{n} \sum_{k=2}^{n} \frac{1}{\log(k)} = 0 .$$
$$\lim_{n\to\infty}\frac{1}{n}\sum_{k=2}^{n}\frac{1}{\log(k)}= 0$$

## A hint to a problem for convergence in probability: $\lim_{n \to \infty} \frac{1}{n} \sum_{k=2}^{n} \frac{1}{\log(k)} = 0$ - Answer 1

https://math.stackexchange.com/questions/1977235
I would like a hint to show that $$\lim_{n \to \infty} \frac{1}{n} \sum_{k=2}^{n} \frac{1}{\log(k)} = 0 .$$
$$\lim_{n\to\infty}\frac{1}{n}\sum_{k=2}^{n}\frac{1}{\log(k)}= 0$$

## A hint to a problem for convergence in probability: $\lim_{n \to \infty} \frac{1}{n} \sum_{k=2}^{n} \frac{1}{\log(k)} = 0$ - Answer 2

https://math.stackexchange.com/questions/1978639
I would like a hint to show that $$\lim_{n \to \infty} \frac{1}{n} \sum_{k=2}^{n} \frac{1}{\log(k)} = 0 .$$
\begin{align}\lim_{n\to\infty}\frac1n\sum_{k=2}^n\frac{1}{\log(k)}& =\lim_{n\to\infty}\left(\frac{\sum_{k=2}^{n+1}\frac{1}{\log(k)}-\sum_{k=2}^n\frac{1}{\log(k)}}{(n+1)-n}\right)\\\\ & =\lim_{n\to\infty}\frac{1}{\log(n+1)}\\\\ & =0\end{align}

## When $\frac 1 n \sum^n_{k=1} a_k \to 0 \implies \sum \frac 1 k a_k<+\infty$ - Answer 1

https://math.stackexchange.com/questions/1720494
Consider the statement $$\lim_{n \to \infty}\frac 1 n \sum^n_{k=1} a_k=0 \implies \sum^{+\infty}_{k=1} \frac 1 k a_k<+\infty$$ Is the statement true for $a_k$ such that $\left|{\frac 1 n \sum^n_{k=1} a_k}\right|<\frac 1 {n^\alpha}$ and $\alpha>0$? Is the statement false in general? (This is not homework, I think I solved this, it is a nice exercise and I would like to have a feedbac...
$$\lim_{n\to\infty}\frac{1}{\ln(\ln n)}\sum_{k=2}^n\frac{1}{k\ln k}$$

## Help with the limit $\lim_{n\to\infty}\frac{1}{\ln(\ln n)}\sum_{k=2}^n\frac{1}{k\ln k}$, most probably with Stolz-Cesaro theorem - Question

https://math.stackexchange.com/questions/2115451
$$\lim_{n\to \infty}\frac{\frac{1}{2\ln2}+\frac{1}{3\ln3}+\ldots+\frac{1}{n\,\ln\,n}}{\ln(\ln\,n)}$$ The result is $1$ (according to the book, though it does not show the steps, which I'm interested in). I've applied the theorem and it led me to an equally unhelpful limit.
$$\lim_{n\to\infty}\frac{1}{\ln(\ln n)}\sum_{k=2}^n\frac{1}{k\ln k}$$

## Help with the limit $\lim_{n\to\infty}\frac{1}{\ln(\ln n)}\sum_{k=2}^n\frac{1}{k\ln k}$, most probably with Stolz-Cesaro theorem - Answer 1

https://math.stackexchange.com/questions/2115503
$$\lim_{n\to \infty}\frac{\frac{1}{2\ln2}+\frac{1}{3\ln3}+\ldots+\frac{1}{n\,\ln\,n}}{\ln(\ln\,n)}$$ The result is $1$ (according to the book, though it does not show the steps, which I'm interested in). I've applied the theorem and it led me to an equally unhelpful limit.
$$\lim_{n\to\infty}\frac{1}{\ln(\ln n)}\sum_{k=2}^n\frac{1}{k\ln k}$$

## Help with the limit $\lim_{n\to\infty}\frac{1}{\ln(\ln n)}\sum_{k=2}^n\frac{1}{k\ln k}$, most probably with Stolz-Cesaro theorem - Answer 2

https://math.stackexchange.com/questions/2115564
$$\lim_{n\to \infty}\frac{\frac{1}{2\ln2}+\frac{1}{3\ln3}+\ldots+\frac{1}{n\,\ln\,n}}{\ln(\ln\,n)}$$ The result is $1$ (according to the book, though it does not show the steps, which I'm interested in). I've applied the theorem and it led me to an equally unhelpful limit.
$$\lim\limits_{n\to\infty}\frac{1}{\ln (\ln n)}\sum\limits_{k=2}^{n}\frac{1}{k\ln k}$$

## Compute $\lim\limits _{n\to \infty }\frac{1}{\ln (\ln n)}\sum\limits_{k=2}^{n} \frac{1}{k\ln k}$ without Taylor series - Question

https://math.stackexchange.com/questions/2244650
Evaluate $$\lim _{n\to \infty }\frac{\sum_{k=2}^{n} \frac{1}{k\ln k}}{\ln (\ln n)}$$ without Taylor series. I applied Stolz–Cesàro's theorem: $$\lim _{n\to \infty }\frac{\sum_{k=2}^{n} \frac{1}{k\ln k}}{\ln (\ln n)} = \lim _{n\to \infty }\frac{\sum_{k=2}^{n+1} \frac{1}{k\ln k} - \sum_{k=2}^{n} \frac{1}{k\ln k}}{\ln (\ln (n+1)) - \ln (\ln n)} = \lim _{n\to \infty } \frac{\frac{1}{(n+1) \ln(n+1)}}...$$\lim\limits_{n\to\infty}\frac{1}{\ln (\ln n)}\sum\limits_{k=2}^{n}\frac{1}{k\ln k}$$## Compute \lim\limits _{n\to \infty }\frac{1}{\ln (\ln n)}\sum\limits_{k=2}^{n} \frac{1}{k\ln k} without Taylor series - Answer 1 https://math.stackexchange.com/questions/2244668 Evaluate$$\lim _{n\to \infty }\frac{\sum_{k=2}^{n} \frac{1}{k\ln k}}{\ln (\ln n)}$$without Taylor series. I applied Stolz–Cesàro's theorem:$$\lim _{n\to \infty }\frac{\sum_{k=2}^{n} \frac{1}{k\ln k}}{\ln (\ln n)} = \lim _{n\to \infty }\frac{\sum_{k=2}^{n+1} \frac{1}{k\ln k} - \sum_{k=2}^{n} \frac{1}{k\ln k}}{\ln (\ln (n+1)) - \ln (\ln n)} = \lim _{n\to \infty } \frac{\frac{1}{(n+1) \ln(n+1)}}...
$$\lim\limits_{n\to\infty}\frac{1}{\ln (\ln n)}\sum\limits_{k=2}^{n}\frac{1}{k\ln k}$$

## Compute $\lim\limits _{n\to \infty }\frac{1}{\ln (\ln n)}\sum\limits_{k=2}^{n} \frac{1}{k\ln k}$ without Taylor series - Answer 2

https://math.stackexchange.com/questions/2244730
Evaluate $$\lim _{n\to \infty }\frac{\sum_{k=2}^{n} \frac{1}{k\ln k}}{\ln (\ln n)}$$ without Taylor series. I applied Stolz–Cesàro's theorem: \lim _{n\to \infty }\frac{\sum_{k=2}^{n} \frac{1}{k\ln k}}{\ln (\ln n)} = \lim _{n\to \infty }\frac{\sum_{k=2}^{n+1} \frac{1}{k\ln k} - \sum_{k=2}^{n} \frac{1}{k\ln k}}{\ln (\ln (n+1)) - \ln (\ln n)} = \lim _{n\to \infty } \frac{\frac{1}{(n+1) \ln(n+1)}}...