Linked Questions

8
votes
3answers
300 views

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

Considering the sum as a Riemann sum, evaluate $$\lim_{n\to\infty}\sum_{k=1}^{n}\frac{k}{n^2+k^2} .$$
5
votes
3answers
186 views

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.
2
votes
2answers
3k views

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 ...
4
votes
1answer
235 views

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 ...
2
votes
2answers
82 views

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?
3
votes
2answers
94 views

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.
4
votes
1answer
169 views

limit of summation [duplicate]

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 n$\...
1
vote
2answers
87 views

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

Find $\lim_{n\to \infty} \sum_{k=1}^n \frac{k}{n^2+k^2}$ Since $\frac{k}{n^2+k^2}\leq \frac{k}{k^2+k^2}=\frac{1}{2k}$, then $\sum_{k=1}^n \frac{k}{n^2+k^2}\leq \sum_{k=1}^n \frac{1}{2k}=\frac{1}{2} \...
1
vote
1answer
62 views

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.
1
vote
1answer
77 views

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 ...
2
votes
1answer
51 views

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? $\lim_{n\rightarrow\infty} \sum_{x = 1}^{n}{\frac{x}{n^2+x^2} }.$
13
votes
1answer
1k views

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.
2
votes
2answers
88 views

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 ...
0
votes
2answers
77 views

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 ...
0
votes
1answer
35 views

Convergence test for partial sum whose elements all change as the index increases

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 ...