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### Evaluate $\lim_{n\to\infty}\sum_{k=1}^{n}\frac{k}{n^2+k^2}$ [duplicate]

Considering the sum as a Riemann sum, evaluate $$\lim_{n\to\infty}\sum_{k=1}^{n}\frac{k}{n^2+k^2} .$$
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### limit $\lim_{n\to \infty }\sum_{k=1}^{n}\frac{k}{k^2+n^2}$ [duplicate]

How do I evaluate this? $$\lim_{n\to \infty }\sum_{k=1}^{n}\frac{k}{k^2+n^2}$$ I got concerned for that, I've tried make it integral for Riemann but it still undone.
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### Find the limit by expressing it as a definite integral of an appropiate function via Riemann Sums. [duplicate]

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 of ...
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### Proving a limit summation [duplicate]

Show that $$\lim_{n \to \infty} \sum_{k=0}^{2n} \frac{k}{k^2+n^2} = \frac{1}{2}\, \log 5$$ How would you prove this? I understand limits, but summations not so much. Would I need to take the ...
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### Show convergence and calculate the limit: $\lim \limits_{n \to \infty}\sum \limits_{k=n}^{2n}\frac{k}{k^2+n^2}$ [duplicate]

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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### Evaluation of the limit $\lim_{n\to\infty}\sum_{k=1}^n\frac k{k^2+n^2}$ [duplicate]

$$\lim_{n\to\infty}\sum_{k=1}^n\frac k{k^2+n^2}$$ I need to evaluate this limit, but I don't know how to start. Should i take $1/n^2$ out? Help required. Thank you.
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### Finding the limit of a series. [duplicate]

I haven't touched series and sequences in a while and am rusty, so need some direction in finding the following: $$\lim\limits_{n \to \infty} \sum_{k=1}^n \frac{k}{n^2 + k^2}$$ Thanks in advance.
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### How to calculate the value of the limit of a sum: $\lim_{n\rightarrow\infty}\sum_{k=0}^{n}\frac{k}{k^{2}+n^{2}}$? [duplicate]

Need to calculate the value of the following limit $$\lim_{n\rightarrow\infty}\sum_{k=0}^{n}\frac{k}{k^{2}+n^{2}}$$ I don't know how. In general would know how to make that kind of limits with ...
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### Calculating $\lim_{n\rightarrow\infty} \sum_{x = 1}^{n}{\frac{x}{n^2+x^2} }$ [duplicate]

Is there a simple method of calculating this limit? $\lim_{n\rightarrow\infty} \sum_{x = 1}^{n}{\frac{x}{n^2+x^2} }.$
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### The limit of a sum $\sum_{k=1}^n \frac{n}{n^2+k^2}$ [closed]

How to compute this limit: $$\lim_{n\to\infty}\left(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\cdots+\frac{n}{n^2+n^2}\right)$$ Please give me some hint.
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### To show that the limit of the sequence $\sum\limits_{k=1}^n \frac{n}{n^2+k^2}$ is $\frac{\pi}{4}$ [duplicate]

Show that $$\lim_{n \to \infty} \sum\limits_{k=1}^n \frac{n}{n^2+k^2} = \frac{\pi}{4}.$$ I am familiar with Taylor series and Fourier series of the standard functions. I tried to compare with those ...
### Value of $\lim_{n\to\infty}\sum_{k=1}^n \frac{n}{n^2+k^2}$ [duplicate]
$$\lim_{n\to\infty}\sum_{k=1}^n \frac{n}{n^2+k^2}$$ I tried this using powerseries just putting as $x=1$ there ,even tried thinking subtracting $s_{n+1} - s_{n}$ would be of some help, also I thought ...
I have a sequence of length $n$, $\{b_i(n),\, i=1,...,n\}$. Each $b_i(n)$ is a strictly positive rational number. I write $b_i(n)$ because indeed each element of the sequence is a function also of ...