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
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Solving a finite sum involving spectral methods

I'm trying to get through Exercise 6.2 of the book Spectral Methods in MATLAB by Nick Trefethen. Exercise 6.1 gives the coefficients of the N+1 by N+1 matrix (indexed from $0$ to $N$) by: $$D_{ij} = \frac{1}{a_j} \prod^N_{\substack{k=0\\k\neq…
Benjamin Longaker
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Proving that $\sum_{k=0}^{n-1} (n-k)(k+1) = \frac{n(n+1)(n+2)}{6}$

Can you give me any hint how to prove that $\sum_{k=0}^{n-1} (n-k)(k+1) = \frac{n(n+1)(n+2)}{6}$ I tried to divide this sum into two cases when $n$ is odd and even but it does not give me any proper result. I can see that the result on the right…
John Snow
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Finding the value of the given summation.

How do I find the value of $f(2022)$ if $$f(n) = \displaystyle \sum_{x=1}^n \dfrac{\sqrt{x}-x}{\sqrt{x+\sqrt{x}}-x}$$ I tried simplifying and rationalizing the denominator but it is not working here. I am stuck on how to proceed with this. Any help…
Alan
  • 409
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evaluate $\sum_{n=0}^{\infty}\frac{x^n}{A(x^{2n}-1)+1}$

i tried a few thing but none them worked out so I'm totally clueless what to do to solve this $$\sum_{n=0}^{\infty}\frac{x^n}{A(x^{2n}-1)+1}$$
mh96
  • 29
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Geometric sum with square root

What is the result of $$\sum_{t=0}^\infty \sqrt t a^t ?$$ where $a \in (0,1)?$ My effort: The geometric sum writes $$\forall a\in (0,1) \qquad \sum_{t=0}^\infty a^t = \frac{1}{1-a}$$ Informally, we can make the derivative of both sides…
Davide Maran
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How to simplify a sum with factorial?

I have the following sum: $$\sum_{n=1}^k \frac{k!}{n!(k-n)!}, \quad k=9$$ wolfram alpha It got simplified to $2^k-1$. How can I do it with math formulas? Thank you!
Chelios
  • 123
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Can a summation be expressed in terms of two variables? How?

Given the following question. What is the sum of all factors of 72? 72 can be expressed as 2^3 x 3^2. The following are the factors of 72. 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72 You simply add all. However I have noticed that you can…
AndroidV11
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How do I prove/disprove this??

I have to do a proof of the statement: $$N(b)\equiv2(2b+1)^2(2b+2)^2=\sum_{k=0}^{2b-1}\Biggl(\prod_{j=1}^4(4b-k+j)\biggl)+\prod_{j=1}^4(2b+j)-(2b+1)\prod_{j=1}^3(2b+j)$$ I know I'm not supposed to ask a question and then have someone solve it, but I…
user1010187
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Finding the sum of $1+4k+9k^2+...+n^2k^{n-1}$

I'm having trouble using Abel's summation formula as $a_1b_1+a_2b_2+...+a_nb_n=(a_1-a_2)(b_1)+(a_2-a_3)(b_1+b_2)+...+(a_{n-1}-a_n)(b_1+b_2+...+b_{n-1})+a_n(b_1+b_2+...+b_n)$ to find the sum of $1+4k+9k^2+...+n^2k^{n-1}$. I know…
Pen and Paper
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Understanding why a sum is greater than another

I'm currently going through a research paper and was trying to redo a proof on my side. The paper states : $$ \begin{align} \text{P}[failure] &\le \text{P}[s + 1 \text{ runs of A fail}] \\ &\le \sum_{i \ge s + 1} \binom{2s + 1}{i}…
olirwin
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$x_1,x_2...x_m$ are integers where $-2\leq x_j\leq1$, for all j=1,2,...m and $S_r=\sum_{j=1}^mx_j^r,S_1=22,S_4=184.$ Then $\max(S_3)-\min(S_3)$ is?

$x_1,x_2...x_m$ are integers where $-2\leq x_j\leq1$, for all j=1,2,...m and $S_r=\sum_{j=1}^mx_j^r,S_1=22,S_4=184.$ Then $\max(S_3)-\min(S_3)$ is? My attempt: $$ \begin{align*} S_1&=x_1+x_2+x_3+\ldots…
Tatai
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If $\phi(r) =1+\frac 12 +\frac 13… \frac 1r$ and $\sum_{r=1}^{n} (2r+1)\phi (r) =P(n)\phi(n+1)-Q(n)$. Find $P$ and $Q$.

I tried making a double sum $$\sum_{r=1}^{n} \sum _{k=1}^r \frac{2r+1}{k}$$ But since the final limits aren’t same the changing of orders cannot be used. Can I get a hint on how to solve it?
Aditya
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Why does this summation of ones give this answer?

I saw this in a book and I don't understand it. Suppose we have nonnegative integers $0 = k_0
Carlos Lobato
  • 139
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Would the following sum give me the following numbers?

Would $\sum_{k=1}^{9}\sum_{j=0}^{9}k*100+10*j+k$ give me the sum of the numbers $101+111+121+131+...+191+202+212+222+...292+303+313+323+...+393...+999$ If not, any advice? Thanks in advance for the help. Edit: +k at the end and j starting at…
Annalisa
  • 911
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What does $\sum_{r=1}^{n} r^{k}$ equal in general

I know that $\sum_{r=1}^{n} r = \frac{n(n+1)}{2}$, $\sum_{r=1}^{n} r^{2} = \frac{n(n+1)(2n+1)}{6}$, $ \sum_{r=1}^{n} r^{3} = \frac{n^2(n+1)^2}{4}$ but what does $\sum_{r=1}^{n} r^{k}$ equal in general ? Can we express it in some simple…
marks_404
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