All Questions
5 questions
3
votes
2
answers
113
views
Convergence of a sum as limit tends to infinity that seems to be harmonic series
I have come across a mathematical problem that is to evaluate the expression:
$$
lim_{n\rightarrow\infty} \left\{\frac{1}{\sqrt{2n-1^2}}+\frac{1}{\sqrt{4n-2^2}}+\frac{1}{\sqrt{6n-3^2}}+...+\frac{1}{\...
- 1,483
0
votes
1
answer
65
views
how to show convergence and evaluate $n\sum_{k=1}^{2n} \frac{e^{-n/k}}{k^2}$
I tried this :
$$\lim\limits_{n \rightarrow +\infty}n\sum_{k=1}^{2n} \frac{e^{-n/k}}{k^2} = \int_{0}^{1} \frac{e^{-1/x}}{x^2} dx$$ to evaluate, and to show convergence, I think $$n\sum_{k=1}^{2n} \...
- 51
1
vote
2
answers
160
views
Finding $\lim_{n\to \infty}\left((1+\frac{1}{n^2})(1+\frac{2^2}{n^2})^2\cdots(1+\frac{n^2}{n^2})^n\right)^{\frac{1}{n}}$
Calculate the following limit:
$$\lim_{n\to \infty}\left((1+\frac{1}{n^2})(1+\frac{2^2}{n^2})^2\cdots(1+\frac{n^2}{n^2})^n\right)^{\frac{1}{n}}$$
My attempt:
Let
$$y=\lim_{n\to \infty}\left((1+\frac{...
- 1,578
0
votes
1
answer
42
views
Convergence of $P_{n}=\frac{1}{n^{2}}\prod_{k=1}^{n}{(n^{2}+k^{2})^{\frac{1}{n }}}$
I got some issues with the convergence of Pn
$P_{n}=\frac{1}{n^{2}}\prod_{k=1}^{n}{(n^{2}+k^{2})^{\frac{1}{n }}}$ , with n⩾1
and finding the limit of this convergent sequence , thanks in advance .
user625562
2
votes
2
answers
123
views
Trying to evaluate a limit of a sum of cosines and am stuck on something that looks like a Riemann sum,
here's where I am stuck:
$$\lim_{n \to \infty} \sum_{k=0}^{n-1} \cos(kx/n)\frac{1}{n}$$
so...it looks like at this point I could convert to a Riemann integral, but to which one?
Maybe $$\int_0^{\...
- 177