All Questions
13
questions
6
votes
2
answers
502
views
How to perform this sum
I encountered this sum
$$S(N,j)= \frac{2 \sqrt{2}h(-1)^j}{N+1}\cdot\sum _{n=1}^{\frac{N}{2}}
\frac{\sin ^2\left(\frac{\pi j n}{N+1}\right)}{\sqrt{2 h^2+\cos \left(\frac{2 \pi n}{N+1}\right)+1}},$$
...
- 712
1
vote
1
answer
52
views
A Riemann Sum clarification
I have to find the area under the curve $f(x) = x^2$ in the segment $[-2, 1]$ using Riemann Sum. So here is what I did, but it's wrong and I need some clarification about (see the questions in the end)...
- 3,482
2
votes
1
answer
59
views
Integration with Riemann Sum
I wanted to try to perform a Riemann Sum for the following integral, but I got stuck in the middle.
$$\int_{-1}^0 e^{-x^2}\ \text{d}x$$
So the interval is $[-1, 0]$, and I chose $\Delta x = \dfrac{1}{...
- 3,482
0
votes
0
answers
175
views
Prove zeta function has coninuous derivatives of all orders [duplicate]
I need to prove that the $\zeta$ function $$\zeta(x):=\sum^{\infty}_{k=1}\frac{1}{k^x}$$defines a function that has continuous derivatives of all orders.
Here's my attempt:
So, the $n$'th derivative ...
user634512
0
votes
2
answers
38
views
Simplifying the upper and lower sum of a Riemann Integral
I have been working on solving a sum for a few hours now but I am not getting the answer on my answer sheet (which is $\frac{46}{3}$).
I have two sums:
$16(\sum^n_{i=1} \frac{i^2-2i+1}{n^3}+\frac{i-...
- 353
0
votes
1
answer
80
views
Riemann sums over dense countable sets
Let $f$ and $g$ be positive, smooth and integrable functions in $\mathbb{R}$, whose derivatives are also integrable.
Assume as well that the expression
$$
\frac{\sum_{q\in \mathbb{Q}} f(q)}{\sum_{q\in ...
- 1,859
1
vote
2
answers
130
views
How to show $\lim_{n\to\infty}n\left\{\sum_{k=1}^n\frac{1}{(n+k)^2}\right\}=\frac{1}{2}$ [duplicate]
Show that $$\lim_{n\to\infty}n\Bigg\{\dfrac{1}{(n+1)^2}+\dfrac{1}{(n+2)^2}+\dfrac{1}{(n+3)^2}+\cdots+\dfrac{1}{(2n)^2}\Bigg\}=\dfrac{1}{2}.$$
Proof:
We can rewrite $$\lim_{n\to\infty}n\Bigg\{\...
- 353
0
votes
1
answer
1k
views
Positive and negative portions of conditionally convergent series
I want to prove that for a conditionally convergent series $\sum_{n = 1}^{\infty} a_n$, that its positive subseries (Let $K := \{k \vert a_k > 0\}$, and the positive subseries is $\sum_{j \in K} ...
- 811
2
votes
3
answers
94
views
Finite Sum evaluation
Can anyone help me in proving the following identity: For any $k\in\mathbb{N}$,
$$
\sum_{n=0}^{k}\frac{(-1)^n(k-n)^{2k}}{n!(2k-n)!}=\frac{1}{2}.
$$
Thank You very much in advance.
- 713
1
vote
1
answer
3k
views
Finding the lower and upper sums of $f(x)=2x+1$
I'm trying to find the upper and lower sums for $f(x)=2x+1$ on the interval $[1,4]$.
$P_6$ is the partition of $[1,4]$ consisting of 7 equally space points, $1, \frac{3}{2}, 2, \frac{5}{2},3,\frac{7}{...
- 3,763
2
votes
2
answers
689
views
How to recognize / convert a tricky limit of an infinite series as a Riemann integral?
Edit: I've modified the sums and integrals below into convergent sums and integrals, but my questions are still the same - how can I convert sums into integrals legitimately? As far as I know, the ...
- 1
2
votes
2
answers
79
views
On sums and identities
I am given the following problem set:
(a) the Riemann $\zeta$-function for $s > 1$ is defined through the convergent sum: $$\zeta(s) := \sum_{n = 1}^{\infty} \frac{1}{n^s}$$
show the identity $...
- 306
12
votes
3
answers
271
views
How prove this limit $\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{i+j}{i^2+j^2}=\frac{\pi}{2}+\ln{2}$
show that: this limit
$$I=\lim_{n\to\infty}\dfrac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}\dfrac{i+j}{i^2+j^2}=\dfrac{\pi}{2}+\ln{2}$$
My try:
$$I=\lim_{n\to\infty}\dfrac{1}{n^2}\sum_{i=1}^{n}\sum_{j=1}^{n}...
user94270