I don't know how to solve the following:  Let $(x_n)_{n=1}^\infty$ be a sequence with property $\sum_{n=1}^\infty |\frac{x_n}{n}|&lt;\infty$. Find a limit $\lim_{n\to\infty}\frac{x_1+x_2+\dots+x_n}{n}$.  Any hint is welcome. Thanks in advance.
124270
Suppose that the series $\sum \limits_{n=1}^{\infty}\dfrac{a_n}{n^{\sigma}} \quad(\sigma&gt;0)$ converges. Prove that $\lim \limits_{n\to \infty}\dfrac{a_1+a_2+\dots+a_n}{n^{\sigma}}=0$.  I applied the Abel transformation, but unsuccessfully.
24947
When $\lim_{n\to\infty} x_n = a$, and $\lim_{n\to\infty} y_n = b$, find the limit, $$\lim_{n\to\infty} \frac{x_1 y_n + x_2 y_{n-1} + \cdots + x_n y_1}{n}.$$ Thank you for your help in advance.
113177
assume  $$x_n=\frac{n+1}{2^{n+1}}\sum_{k=1}^n\frac{2^k}{k} ,n=1,2,.....$$  how compute $\lim_{n\to +\infty}x_n$?  Thanks for any hint
68075
Problem: Find with proof limit of the sequence $a_n=1+\frac{1}{a_{n-1}}$ with  $a_1=1$  or show that the limit does not exist.  My attempt:  I have failed to determine the existence. However if the limit exists then it is easy to find it.  The sequence is not  monotonic and I have failed to find any monotonic subsequence subsequence. The sequence is clearly bounded below by $1$. I have observed that the sequence is a  continued fraction so it alternatively increases and decreases.   So, please help me.
52332
I have a question which is giving me a hard time.    I want to show that $$ \lim_{n\to \infty}\frac{1}{n} \sum_{k=1}^n x_k =x$$ given that $\lim_{n\to \infty} x_n= x$.
545
Let $s_n$ be a sequence defined as given below for $n \geq 1$. Then find out $\lim\limits_{n \to \infty} s_n$.   \begin{align}   s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx  \end{align}  I have written a solution of my own, but I would like to know it is completely correct, and also I would like if people post more alternative solutions.
87671
How to use the formal definition of the limit of a sequence to prove the following limit?  $\lim_{x \to -2} {(x^3 + 4x^2+ 4x-1)} = -1,$  $\lim_{x\to 1}{(\frac{1}{x})} = 1.$
81542
I'm doing exercises. In the related book, there is a claim. Is this right? I'm not sure.     For a sequence $\{a_n\}$, there exists a limit $a$ such that $\lim_{n\to\infty} a_n=a$ if and only if for any $p\in \Bbb N$, $\lim_{n\to\infty} |a_{n+p}-a_n|=0$   If not, could you kindly give some counterexamples?  Thanks.
121419
How to prove the following: $$ \lim_{n \to \infty} \int_0^1 \int_0^1 \cdots \int_0^1 \frac{x_1^2+x_2^2+ \cdots +x_n^2}{x_1+x_2+ \cdots +x_n} dx_1 dx_2 \cdots dx_n = \frac23 $$  I would really appreciate if you could help me!
82086
Find the following limit:  $$\lim_{n\to \infty} (1+n)^{\frac1n}$$  I know that  $\lim_{n\to 0} (1+n)^{\frac1n}= e$.
61993
What is the limit of:  $$\lim_{n\to \infty} \frac{n^1+\dots+n^n}{1^n+\dots+n^n} = ?$$  I did some computation with big numbers, I guess it is in the interval $\left(\frac{1}{2},1\right)$.
44415
Let $\sum a_n$ be a convergent series, and let $S = \lim s_n$, where $s_n$ is the nth partial sum.   I need to prove the following:    $\lim_{n \to \infty} \frac{s_1+...+s_n}{n} = S$    How do I go about proving that proof?  Definition of a limit   $\lim_{n \to \infty} f(x) = L$ if for every number $\epsilon&gt;0$ there is some number $\sigma &gt;0$ such that $|f(x)-L| &lt;\epsilon$ whenever $0&lt;|x-a|&lt;\sigma$
111865
Problem: Let $(x_i)$ be a sequence in (0,1) such that $\lim_{n \to \infty} (1/n)\sum_{i=1}^n x_i^k$ exists for every $k=0,1,2,...$ and $f\in C[0,1]$. Show that $\lim_{n \to \infty} (1/n)\sum_{i=1}^n f(x_i)$ exists.  I've used the fact that $B_n(f)$ converges uniformly to $f$ to get   $\lim_{n \to \infty} (1/n)\sum_{i=1}^n f(x_i) = \lim_{n \to \infty} (1/n)\sum_{i=1}^n \sum_{k=0}^n f(\frac{k}{n}) {n \choose_k }x_i^k (1-x_i)^{n-k}$  And then you can arrange to get   $\lim_{n \to \infty} (1/n) \sum_{k=0}^n \left[ f(\frac{k}{n}) {n \choose_k }\sum_{i=1}^nx_i^k (1-x_i)^{n-k}\right]$  I want to use the fact that $(1-x_i) &lt; 1$ to consolidate some of the equation but I get nowhere.
25796
If $$\left\{x_{n}\right\}\mid x_{1}=5,x_{n+1}=x_{n}^{2}-2,\forall n\geq 1$$ find $$\lim_{n\to\infty}\frac{x_{n+1}}{x_{1}x_{2}\cdots x_{n}}.$$  If someone could help me out with tags, it'd be lovely. I think this is calculus and real-analysis, but I'm not sure--I had the problem scribbled down on a post-it, and I forget where it's from.
124275
Let $\{a_n\}$ be a sequence of real numbers such that $$\lim_{n \to \infty}\left|a_n+ 3\left(\frac{n-2}{n}\right)^n \right|^{\frac{1}{n}}=\frac{3}{5}.$$  What is the value of $\lim_{n\to \infty} a_n$?
64874
Let $m \ge 2 -$ fixed positive integer. The sequence of non-negative real numbers $\{x_n\}_{n=1}^{\infty}$ is that for all $n\in \mathbb N$   $$x_{n+m}\le \frac{x_n+x_{n+1}+\cdots+x_{n+m-1}}{m}$$   Prove that this sequence has a limit.   I know I need to prove 1) the sequence is monotone; 2) the sequence is limited. But I can not do any of these items.
96663
Need to find $\lim\limits_{n\to\infty}(a_1^n+\dots+a_k^n)^{1\over n}$ Where $a_1\ge\dots\ge a_k\ge 0$  I thought about Cauchy Theorem on limit $\lim\limits_{n\to\infty}\dfrac{a_1+\dots+a_n}{n}=\lim a_n$ and something like what happen if all $a_i=0$ or $a_1=\dots=a_k$, but may be something I am thinking wrong?  Maybe it is too simple but I am not getting it; please help.