Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [summation]

Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.

The notation $\sum\limits_{i=1}^na_i$ means $a_1+\ldots +a_n$.

Use for sums of infinite series and questions of convergence; use for questions about finite sums and simplification of expressions involving sums.

17770 questions
2
votes
1 answer

How does a multiindex work?

Can anyone explain me how a multiindex works? For example here: \begin{equation}\begin{aligned} &|\overline{r}|= r_0 + r_1 + \cdots + r_n\\ &\sum_{j=1}^{m}\sum_{|\overline{r}|=\mu_j}a_{ij}\left(\frac{\partial \tau}{\partial{x_0}} \right)^{r_0}…
mathmaniac
  • 534
2
votes
1 answer

Find the sum of the following for any positive integer n where $\langle n \rangle$ denotes the integer nearest to $\sqrt{n}$

Show that $$\sum_{n=1}^{\infty} \dfrac{2^{\langle n \rangle}+2^{-\langle n \rangle}}{2^n} = 3$$ I think I can do the following which I am not quite sure about: $$\sum_{n=1}^{\infty} \dfrac{2^{\langle n \rangle}+2^{-\langle n \rangle}}{2^n}…
Saradamani
  • 1,579
2
votes
1 answer

Geometric summation

The sum $a_1+a_2+a_3+...+a_n$ is geometric. If $a_1+a_3+a_5=455$ and $a_2+a_4+a_6=1365$, then the ratio between each consecutive term is $2$, $3$,$4$ or an other number? Answer is supposed to be $3$. My progress: Since this is a geometric sum we can…
EricAm
  • 1,070
2
votes
1 answer

Sum of $\sum_{n= 1}^{\infty} \frac{(-1)^n \ln(n)}{n(n+1)}$

I am curious about this sum because (as wolfram alpha tells me) it simplifies to a rational number: $$\sum_{n= 1}^{\infty} \frac{(-1)^n\ln(n)}{n(n+1)} = 0.063254$$ I found this interesting because I did not expect this complicated sum to converge so…
user3760593
  • 319
2
votes
0 answers

$\sum_{n=0}^{\infty}\frac{\sin(2n+1)x}{2n+1}$

I need to find $$\sum_{n=0}^{\infty}\frac{\sin(2n+1)x}{2n+1}$$ I know that if I find a appropriate function and expand it as a Fouries series, I can find the sum. But it seems to me like way with a hindsight. I don't know the way to find the…
Septacle
  • 461
2
votes
1 answer

Sum of $ \sum_{n=2}^{\infty}(-1)^n *\frac{ \ln(n-1)}{n}$

I've seen a similar question to this on this site: Calculate sum of $\sum_{n=1}^{\infty}(-1)^n\frac{\ln n}{n}$. However, this sum is a little different because the argument within the natural log function does not match the denominator and the sum…
user3760593
  • 319
2
votes
3 answers

How to calculate sum of the integers from $m$ to $n$.

How to calculate the sum of the integers from $m$ to $n$? Is this correct? $$ \frac{n (n+1)}{2} - \frac{m (m+1)}{2}$$
Saurabh Chandra Patel
  • 131
2
votes
4 answers

How to explicit the summation

I have the following summations: $$ \sum_{i = 0}^{n-1} \sum_{j = 0}^{n-2} \sum_{k =j+1}^{n-1} 1 $$ and I know that the first step should be like this: $$ \sum_{i = 0}^{n-1} \sum_{j = 0}^{n-2} (n - j - 1)$$ But I don't know how to get this. What is…
emaph
  • 23
2
votes
5 answers

Problem With Sum

I would like help with the following summation. Every time I come to this step in my signals class, I have difficulty proving that: $$ \sum_{j=0}^{n-3}1=n-2 $$
Serkan Yassıkaya
  • 21
  • 1
2
votes
3 answers

Prove a simple property of summation

I've come across the following fact many times, and I used to take it for granted. However, I cannot think of a both beautiful and rigorous proof of it: Fact 1:For two non-empty finite sets $A,B$ and map $f:A\times B\to \mathbb R,$ If we have …
painday
  • 968
2
votes
1 answer

Calculate sum in a short way

The question is like this: Let the sum $$\sum_{n=1}^9 \frac{1}{n(n+1)(n+2)}$$ written in its lowest terms be $\frac{p}{q}$. Find $p-q$ I tried to calculate it by putting in values 1 to 9 and actually calculating the value of the sum, but it was too…
user167920
  • 427
2
votes
2 answers

convergence in distribution of a sum of random variables

Consider $X_i$'s as iid random variables with mean 0 that are not point mass (they are non-degenerate) and they have finite variance. $a_i$'s are constants that are finite which converge to $0$. How can I show that that $\sum_{i = 1}^{n} a_iX_i…
user48405
2
votes
1 answer

Rearranging A Double Summation

I have the following: $$\sum_{i=0}^n \sum_{k=0}^i a_i {i \choose k} (-b)^{i-k} x^k$$ I want to find a way to express $x^k$ in terms of the outer summation so it would be $x^i$. Is there any way to do this?
Feltie
  • 35
2
votes
2 answers

Find a formula for $\sum\limits_{n=0}^N(-1)^n\frac{(2n+1)^3}{(2n+1)^4+4}$

I found this sum in the mathematial induction chapter of The art of Computer Programming and i have no idea how to solve it. $\dfrac{1^3}{1^4+4}-\dfrac{3^3}{3^4+4} + ... +\dfrac{(-1)^n(2n+1)^3}{(2n+1)^4+4} $ I tried writing it as…
Andrei Mihailescu
  • 33
2
votes
5 answers

Is $\sum_{k=1}^n\sum_{j=1}^nk\cdot j=\sum_{k=1}^nk\cdot\sum_{j=1}^nj$?

I have found confronting information on other websites and thus will ask my question here. Is the following conversion legitimate? $$\sum_{k=1}^n\sum_{j=1}^nk\cdot j=\sum_{k=1}^nk\cdot\sum_{j=1}^nj$$ I have seen someone doing something similar on…
Philipp
  • 259
1 2 3
…
99 100