All Questions
47
questions
0
votes
1
answer
77
views
Please help me to find the sum of an infinite series. [duplicate]
Please help me to solve this problem. I need to find the sum of an infinite series:
$$
S = 1 + 1 + \frac34 + \frac12 + \frac5{16} + \cdots
$$
I tried to imagine this series as a derivative of a ...
- 21
0
votes
1
answer
117
views
Minimising problem with sums and matrices
We assume that $p_0,...,p_k$ are pairwise conjugate to Q and we define a function $f(x)=\frac{1}{2}x^TQx+g^Tx$, and then we have to show that the solution $\alpha_0,...,\alpha_k$ is $\alpha_k=-\frac{...
- 558
1
vote
1
answer
94
views
Finite sum of csc via sum of cot [closed]
Would anyone give a clue how to prove the following identity:
$\sum_{k =1}^{n-1} \csc\left(\frac{k\pi}{n}\right) = -\frac{1}{n}\sum_{k=0}^{n-1}(2k+1)\cot\left(\frac{(2k+1)\pi}{2n}\right)$.
I tried ...
- 19
0
votes
1
answer
145
views
Calculating an infinite sum and finding a limit
Is it possible to evaluate or find the solution to the following infinite sum with an inverse trigonometric function (arctan), and its limit?
$\displaystyle\lim_{x\to\infty}\frac{1}{\sqrt x}\sum_{n=1}^...
- 293
1
vote
1
answer
98
views
Upper bound on the integral of a step function inequality
Problem
I have been working through content in real analysis and came across across an inequality in a proof that is unclear to me.
We are told that a function $\psi$ is a step function on the ...
- 4,345
3
votes
2
answers
221
views
Convergence of $\sum_{n=0}^{\infty} \frac{4^n}{3^n+7^n}$
I need help with this.
$\sum_{n=0}^{\infty} \frac{4^n}{3^n+7^n}$
I know that it converges but i can not proove why.
I tried to rewrite it, it seems to be a geometric serie. I tried to do a common ...
- 47
1
vote
0
answers
343
views
How to find tight upper bound?
I am trying to find a tight upper bound on $\sum_{i=1}^nf(s_i)\left(\frac{1 - \exp(-s_it)}{s_it} - \frac{1}{1 + s_it}\right)$ where the maximum value of function is $a$ and minimum is $b$, and both of ...
- 81
0
votes
1
answer
462
views
The Heaviside functions in the following?
For $(\mu_{i})_{i=\overline{1,n}}$ are real positive parameters, we have $H$ is the Heaviside function, i.e
$$\forall i = \overline{1,n}, ~~~~H(u-\mu_i)=\left\{\begin{array}{ll} 1 & \quad \mbox{if ...
- 37
1
vote
1
answer
116
views
How do I obtain the Jacobi theta function?
If I enter the series $$\sum_{n=-\infty}^\infty e^{-(n+x)^2}$$ into WolframAlpha it expresses it by the Jacobi theta function $$\sqrt{\pi} \vartheta_3(\pi x,e^{-\pi^2})=\sqrt{\pi} \big(1+2\sum_{n=1}^\...
- 139
0
votes
1
answer
66
views
Closed form of this series
I am wondering if the following series has a closed form expression:
$\sum_{x=0}^{\infty} \frac{g(x)\lambda^x}{x!}$, where $g(x)=\frac{x}{2}$ for even $x$, and $g(x)=\frac{x-1}{2}$ for odd $x$.
I ...
- 284
1
vote
1
answer
53
views
Sum of the absolute values
I was going over a question and need your opinion about solution of the sum of the absolute values as
$$S_n = |0-a|+|1-a|+|2-a|+ \dots + |(n-1)-a|+|n-a|$$
where a is a constant term. What could be ...
- 21
1
vote
2
answers
56
views
Problems with understanding a given Sum Identity
In a textbook I found the following equation without any explaination:
\begin{align}
\sum_{\substack{j, k=1,\\ j\neq k}}^n\frac{1}{(x-x_j)(x-x_k)} - \frac{3}{2}\left( \sum_{j=1}^n \frac{1}{x-x_j}\...
- 974
2
votes
3
answers
53
views
Simple question about a formula for sums
Why is
$$
\left(\sum_{i=1}^{n}a_i\right)^2= \sum_{i=1}^{n}\sum_{j=1}^{n}a_ia_j
$$
I guess it is fairly clear when you do it for $n=2$ for example, but I can't really proof it for all $n$ with ...
- 1,652
0
votes
1
answer
71
views
Finding the infinite sum using Leibniz Test
I have been given a task in my previous lecture to determine whether the infinite sum;
$$\sum_{n=1}^\infty \frac{\cos (\pi n)\ln n}{n}$$
Converges or diverges.
My perspective on the problem is that ...
- 1,126
1
vote
1
answer
49
views
Continous function $r(x)$ on $[0,1]$ such that $\lim_{n\rightarrow\infty}\sum_{k=0}^{n}\frac{1}{n}g(\frac{k}{n},a,b)r(\frac kn)=0$
The problem is related to this question (https://math.stackexchange.com/posts/3073378/edit).
Is there any general way to construct a non-zero continous function $r(x)$ on $[0,1]$, independent of ...
- 1,260