# Tagged Questions

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### How find the limit $\lim_{n\to+\infty}\sum_{i=2}^{n}\dfrac{\ln{i^2}}{i^2}$

find the $$\lim_{n\to+\infty}\left(\dfrac{\ln{2^2}}{2^2}+\dfrac{\ln{3^2}}{3^2}+\dfrac{\ln{4^2}}{4^2}+\cdots+\dfrac{\ln{n^2}}{n^2}\right)$$ My try: ...
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### $\lim_{n\rightarrow\infty}\frac{a_n}{1\cdot 2}+\frac{a_{n-1}}{2\cdot 3}+\ldots +\frac{a_1}{n(n+1)}$

$\displaystyle \lim_{n\rightarrow\infty}\frac{a_n}{1\cdot 2}+\frac{a_{n-1}}{2\cdot 3}+\ldots +\frac{a_1}{n(n+1)}$ if $\lim_{n\rightarrow\infty}a_n=a$ Note that ...
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### $\lim_{n\rightarrow\infty}\sum_{k=1}^n\left(\sqrt{1+\frac{k}{n^2}}-1\right)$

$\displaystyle\lim_{n\rightarrow\infty}\sum_{k=1}^n\left(\sqrt{1+\frac{k}{n^2}}-1\right)$ Note that $\forall x\ge 0, \sqrt{x}-1\le\sqrt{1+x}-1\le x$ Then ...
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### How find this $f(k)=\sum_{n=1}^{\infty}\dfrac{n^k}{2^n}$ is positive integers?

Question: let $$f(k)=\sum_{n=1}^{\infty}\dfrac{n^k}{2^n},k\in N^{+}$$ show that: $f(k)$is always postive integer numbers. this is problem is my creat it,maybe is old problem,because when I ...
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### How many methods to this limit $\lim_{n\to\infty}\left(\dfrac{1}{2}+\dfrac{3}{2^2}+\cdots+\dfrac{2n-1}{2^n}\right)$

find the limit $$\lim_{n\to\infty}\left(\dfrac{1}{2}+\dfrac{3}{2^2}+\cdots+\dfrac{2n-1}{2^n}\right)$$ My try: let $$S=\dfrac{1}{2}+\dfrac{3}{2^2}+\cdots+\dfrac{2n-1}{2^n}$$ ...
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### limit of summation

Using Riemann integrals of suitably functions, find the following limit $$\lim_{n\to \infty}\sum_{k=1}^n \frac{k}{n^2+k^2}$$ Please help me check my method: Let $$f(x)=\frac{x}{1+x^2}$$ For each ...
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### Limit of the sum $\sum_{k=1}^\infty\frac{\sin(kx)}{kx}$

The sum $$S(x)=\sum_{k=1}^\infty\frac{\sin(kx)}{kx}$$ can be written in a closed form: $$S(x)=\frac{1}{x}\left(\frac{1}{2}i\left(\ln(1-\exp(ix)\right)-\ln\left(1-\exp(-ix)\right)\right)$$ I am in ...
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### How to find the value of this sum [closed]

How to simplify this expression and find the value? $\sum_{i=1}^\infty \frac{i}{2^{i-1}}$
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### Does $\lim\limits_{n\to\infty}\frac{\sum_{i=1}^{n^2} 1}{\sum_{i=1}^n i}$ result into 2?

How does the following limit $$\lim\limits_{n\to\infty}\frac{\displaystyle\sum_{i=1}^{n^2} 1}{\displaystyle\sum_{i=1}^n i}$$ result into 2? Something like: ...
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### A limit on binomial coefficients

Let $$x_n=\frac{1}{n^2}\sum_{k=0}^n \ln\left(n\atop k\right).$$ Find the limit of $x_n$. What I can do is just use Stolz formula. But I could not proceed.
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### Help finding the limit of a sum

Hi I'm trying to find the following limit: $$\lim_{n \rightarrow \infty} \frac{1}{n} \sum_{j=1}^{ n } (1 - e^{\frac{-jt}{n}} )$$ expressed as a funciton of t. You may even be able to get it from ...
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### Show convergence and calculate the limit

I guess it has something to do with riemann sums but this is new for me. $\displaystyle\lim \limits_{n \to \infty}\sum \limits_{k=n}^{2n}\frac{k}{k^2+n^2}$ How do i start?
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### Convergence of $\sum_{i=2}^{\infty}\frac{1}{i \cdot f(i)}$

[Edited to fix typo] Is there a precise formulation for when the sum $$\sum_{i=2}^{\infty}\frac{1}{i \cdot f(i)}$$ converges, in terms of the function $f$? Assume that $f$ is smooth and ...
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### Convergence of the series

Im trying to resolve the next exercise: $$\sum_{n=1}^\infty\ e^{an}n^2 \text{ , }a\in R$$ I dont know in which ranges I should separe the a value for resolving the limit and finding out the ...
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### limits and summation [duplicate]

Possible Duplicate: Riemann’s Integrals Question I have the following question, $$\lim_{n\to\infty} \sum_{i=1}^n \tan((\frac \pi {3n})i) \times \frac \pi {3n}$$ and was wondering which ...
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Can someone show that the limit bellow approaches 1/2? Can you also prove that it does, with out using the prime number theorem? $$\lim_{n\to\infty} \frac{\sum\limits_{k=1}^n \Lambda(k) ... 1answer 131 views ### Bounding a Von-Mangoldt Summatory Function Can someone find a function f(n) satisfieing these bounds? Can you also prove that it does?$$ \sum\limits_{k=1}^n \Lambda(k) ...
Find the limit $$\lim_{n\to \infty}\sum_{i=1}^{n}\frac{i}{n^2+i^2}$$ by expressing it as a definite integral of an appropiate function via Riemann Sums. Observation: $n$ must refer to the number ...
I have the following limit: $$\lim_{n\rightarrow\infty}e^{-\alpha\sqrt{n}}\sum_{k=0}^{n-1}2^{-n-k} {{n-1+k}\choose k}\sum_{m=0}^{n-1-k}\frac{(\alpha\sqrt{n})^m}{m!}$$ where $\alpha>0$. ...