Could anyone help me solving the following limit?   $\lim_{n\rightarrow \infty} (\frac{1}{n+1}+ ... +\frac{1}{2n})$  I think it should just be 0 since we can distribute the limite inside the sum, but I am not sure of this answer. Thanks so much for your help!
124283
Show that $$\lim_{n\to \infty}\left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{6n}\right)=\log 6$$ Here I need to use the definition of integral but I  faced problem in range . Please help.
86379
I have to solve: $\lim\limits_{x\to \infty} \frac{\ln(x+1)}{\ln(x)}$. Can you give me any hints to go? Thanks a lot!
121280
Let's $x_n\ge0$ and $$\overline{\lim}_{n\rightarrow\infty}\frac{x_1+x_2+\cdots+x_n}{n}\lt+\infty,~~\lim_{n\rightarrow\infty}\frac{x_n}{n}=0.$$Prove that $$\lim_{n\rightarrow\infty}\frac{x_1^2+x_2^2+\cdots+x_n^2}{n^2}=0.$$ I have no idea how I can start the proof.  Thanks!
19821
I've been trying to solve this limit: $$\lim_{n\to \infty} \left(\cos\left(\frac{1}{n}\right)\right)^{n^2}$$ I solved it using l'hopital but I have to try in another way. I tried to make the expression:  $$ \lim_{n \to \infty}\exp\left(n^2\ln\left(\cos\left(\frac{1}{n}\right)\right)\right)$$  but it didn't work out for me...
92775
Let $\alpha$ be a positive number. Calculate $$\lim_{n\to\infty} \sum_{k=1}^{n} \Big(\frac{k}{n}\Big)^{\alpha k}$$  Edit: I have deleted my attempt, it didn't seem to lead me anywhere and I discovered a couple of mistakes. :)  Please give me a hint!
108186
I'm struggling to find the solution of the following:  $$\lim_{x\rightarrow\infty}(\sqrt{x^2+4x} - x)$$  I come to the answer of $0$.  The book has an answer of $4/1$.  The book explains a part of the question briefly. Looking at the brief answer information given I then come to an answer of $2$...  What is the answer?
105998
I'm trying to study the following series: $$\sum_{n=1}^\infty \frac{n^n}{(n!)^2}$$ All the terms are positive. But I can't solve the limit: $$\lim_{n\rightarrow \infty} \frac{n^n}{(n!)^2}=\lim_{n\rightarrow \infty} \frac{n}{n^2}*\frac{n}{(n-1)^2}*\frac{n}{(n-2)^2}*...*\frac{n}{9}*\frac{n}{4}*n=\lim_{n\rightarrow \infty} \frac{n}{(n-1)^2}*\frac{n}{(n-2)^2}*...*\frac{n}{9}*\frac{n}{4}$$  the first term of the product tends to $\frac{1}{n}$ but I can't say that all the product is $&lt;\frac{1}{n}$ because the other terms  are $&gt;\frac{1}{n}$  how can I solve it?
100702
Find the value of the limit  $$\lim_{n\to \infty} \sum_{j=n}^{4n} {4n \choose j}\left(\dfrac{1}{4}\right)^{j} \left(\dfrac{3}{4}\right)^{4n-j}$$  First we take the limit of the summation from $0$ to $\infty$ then subtract the sum from $0$ to $n-1$. The value of the first sum is $1$. I can't determine the second sum with limit. How we find it?
104005
Let $x_0 = 1, x_1 = \frac{1}{1+x_0}, x_2 = \frac{1}{1+x_1} \cdots x_n = \frac{1}{1+x_{n-1}}$  Find $\lim_{n\rightarrow \infty} x_n$ as $n$ approaches infinity.  I don't know to represent this recurrence relation in a way that I can solve for $x_n$ and take the limit. Is there another way that I can do this?
96491
I have a strong feeling that the following limit is zero, can anybody help me prove it.    $ \lim\limits_{(x,y,z) \to (0,0,0)} \frac{{xyz}}{{x+y+z}}$   Thanks!
29157
Please give me some hints or solution for this limit. I had it on my exam and I'm curious how to solve it. $$ \lim_{x\to 0} \frac{(x+1)^\frac{1}{x}-e}{x} $$
38619
Problem :   Let $a$ be a positive number. Then $$\lim_{n \to \infty}\left[\frac{1}{a+n}+\frac{1}{2a+n}+\cdots +\frac{1}{na+n}\right]$$  Please suggest how to proceed in such limit problems, will be of great help thanks.
84329
Solve for   $$ \lim_{x \rightarrow \infty} \frac{x^{\ln(x)}}{(\ln(x))^x} $$   I used l'Hôpital's rule but in vain. Could you help me someway. Thanks for attention.
117296
Find $$\lim_{n\rightarrow\infty}\frac{1}{\sqrt{n}}\left(\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+\ldots+\frac{1}{\sqrt{2n-1}+\sqrt{2n+1}}\right)$$
48461
How to prove the following limit?   $$\lim_{n \to \infty} (1+1/n)^n = e$$    I can only observe that the limit should be a very large number!   Thanks.
123186
Let $Y_{n} &gt; 0$ for all $ n\in \mathbb{N} $, with $\sum{Y_{n}}= +\infty$.  If $\displaystyle\lim\limits_{n\rightarrow \infty}\frac{X_{n}}{Y_{n}}= a$ then $\displaystyle\lim\limits_{n\rightarrow\infty}\frac{X_{1}+X_{2}+X_{3}+\dots+X_{n}}{Y_{1}+Y_{2}+Y_{3}+\dots+Y_{n}}= a$  I need a idea for solution.  can i supose that $\lim\limits_{n\rightarrow\infty}Y_{n}$ exist ?
109362
Evaluate the limit      $$ \lim_{n\rightarrow \infty}\left(\frac{n+1}{n}\right)^{n^2}\cdot \frac{1}{e^n} $$   My Attempt:  $$ \lim_{n\rightarrow \infty}\left(\frac{n+1}{n}\right)^{n^2}\cdot \frac{1}{e^n} = \lim_{n\rightarrow \infty}\left(1+\frac{1}{n}\right)^{n^2}\cdot \frac{1}{e^n} $$  How can I solve the problem from this point?