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I have a problem with the calculation of the following limit. \begin{equation} \lim_{n\rightarrow \infty} \frac{1+\sqrt{2}+\sqrt[3]{3}+\cdots+\sqrt[n]{n}}{n} \end{equation} I do not know where to start! Thank you very much

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    Do you know the limit of $n^{1/n}$? Also $n=\sum_{i=1}^n 1$.2012-11-19
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    Use $\sqrt[n]{n}\to1$, see e.g. [here](http://math.stackexchange.com/questions/115822/lim-n-to-infty-n-frac1n), and [Cesaro mean](http://math.stackexchange.com/questions/207910/prove-convergence-of-the-sequence-z-1z-2-cdots-z-n-n-of-cesaro-means).2012-11-19
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    As $n$ grows large, what you do when you co from case $n$ to case $n+1$ is adding $1$ to the denominator and adding ever so slightly more than $1$ to the numerator. Intuition dictates it tends towards $1$, and once you have a convergence candidate, you're halfway there.2012-11-19
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    See also: [Evaluating Limit Question $\lim\limits_{n\to \infty}\ \frac{1+\sqrt[2]{2}+\sqrt[3]{3}+\cdots+\sqrt[n]{n}}{n}=1$?](https://math.stackexchange.com/q/130442)2018-04-17

2 Answers 2

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There are at least two possibilities. In all of them you use that $\lim_{n\to\infty}=\sqrt[n]{n}=1$.

The first one uses the following result: if $a_n$ is a convergent sequence, then $$\lim_{n\to\infty}\frac{a_1+a_2+\dots+a_n}{n}=\lim_{n\to\infty}a_n.$$

The second is to use Stolz's criterion.

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    I remember that there's some theorem saying that, given 2 positive sequences $(a_n); (b_n)$ if $\lim \frac{a_n}{b_n} = \alpha$, then $\lim \frac{\mathop\sum_{i = 1}^n a_i}{\mathop\sum_{i = 1}^n b_i} = \alpha$. Is this correct, and what's the name of this theorem?2012-11-19
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    @user49685 It is (a version of) Stolz-Cesaro theorem, see e.g. [this answer](http://math.stackexchange.com/questions/100338/limit-of-quotient-of-two-series/100542#100542).2012-11-19
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Hint: Note that $$\sqrt[n]{n} \rightarrow 1.$$

You'll need to know and use that fact.

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    @DonAntonio I second that.2012-11-19