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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If $\sum_{i = 1}^n a_i b_i = \sum_{i = 1}^n a_i^2 = \sum_{i = 1}^n b_i^2$, then $a_i = b_i$ for all $i$.

I have a statement which I'd have needed to solve a textbook exercice. I found a way around it, but I'm still curious about it - it seems reasonable, but I can't find the right argument for it. Here it is. Let $a_1, \ldots, a_n, b_1, \ldots, b_n$…
Manuel Lafond
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Evaluate $\sum_{i=1}^{25} 2i(i-1)$

Evaluate the sum: $$\sum_{i=1}^{25} 2i(i-1)$$ All I could do is: $$2 \sum_{i=1}^{25} i (i-1)$$ What can I do after this? Is there a way to evaluate without inserting every single integers? Thank you
didgocks
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How can I simplify: $\sum^{n-1}_{i=1}\sum^{n}_{j=i+1}\sum^{j}_{k=1} 1?$

I simplified the most inner sum to: $j$. So, now I have: $$\sum^{n-1}_{i=1}\sum^{n}_{j=i+1}j$$ I'm not sure if the following is correct but here is what I am thinking. I can re-index the inner sum by letting $m = j - i$. So, now I…
Jeel Shah
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Evaluation of Infinite series summation.

For any Positive integer $n\;,$ Let $t(n)$ denote the integer closest to $\sqrt{n}\;,$ Then value of $\displaystyle \sum^{\infty}_{n=1}\frac{2^{t(n)}+2^{-t(n)}}{2^n}$ $\bf{My\; Try::}$ Here What i Understand is that $t(2)=1$ and $t(3) = 3$ So Let…
juantheron
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How to calculate a sum using a geometric series

How to calculate this with a simple calculator. $$\sum_{i=20}^n=59 0.1\cdot600\cdot1.04^{60-i} = \text{ ?}$$ I tried this but it's wrong. Can somebody please tell me where I made a mistake? =0,1*600*1,04^(60-20)*(1-1,04^60-(59-20+1)/1-1,04) With…
alison monroe
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Summation $\frac{3}{1^2}+\frac{5}{1^2+2^2}+\frac{7}{1^2+2^2+3^2}+....$

I came across a question today... Q. The sum $\dfrac{3}{1^2}+\dfrac{5}{1^2+2^2}+\dfrac{7}{1^2+2^2+3^2}+....$ upto $11$ terms is? Okay, I think it can be written as $$\sum_{r=1}^{11}\dfrac{2r+1}{1^2+2^2+...+r^2}$$ I can't see how to simplify…
manshu
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Calculation of sum with many indexes

How to calculated this sum in the closed form? $$ \sum_{(i_1,i_2,\ldots,i_k)\atop 1\leq i_1
Gulmira
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Sum of the Powers of $2$

Suppose I have a sequence consisting of the first, say, $8$ consecutive powers of $2$ also including $1$: $1,2,4,8,16,32,64,128$. Why is it that for example, $1 + 2 + 4 = 7$ is $1$ less than the next term in the series, $8$? Even if one was to try,…
user316321
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Prove expression is positive

Let $\sum_i w_i=1$ and $w_i, x_i \in \mathbb{R}$, show that $$\sum_i w_i x_i^4-\sum_i w_i x_i \sum_i w_i x_i^3\geq 0. $$ I can show that $\sum_i w_i x^2 -\sum_i w_i x_i \sum_i w_i x_i \geq 0$ by convexity of the function $f(y)=y^2$, but i am not…
kiara
  • 213
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Binomial expansion for harmonic numbers

It is well known that harmonic numbers in general cannot be expressed through elementary functions. I am interested in following sums: \begin{equation} {\mathcal S}_\theta(n):=\sum\limits_{p=0}^n \binom{n}{p} (-1)^p…
Przemo
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Mistake in proof for sequence of cubes

I know there are thousands of proofs for this to have a look at, but I started one myself in a slightly different way than what is easily found when googling. To me, the proof seems like it should work, but the result is not what it should…
Mr Tsjolder from codidact
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How do you show $n^3$ as a sum of terms?

We know that $n^2 = \displaystyle\sum_{i=1}^n 2i-1$. Is there a similar way to represent $n^3$ as: $\displaystyle\sum_{i=1}^n ?$, where we replace the question mark with a term?
Mihir Chaturvedi
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How to sum Gaussian function on a grid?

Can anybody help to tell how to sum $$\sum_{-\infty}^{\infty} \sum_{-\infty}^{\infty} e^{-\frac{x^2+y^2}{2}}$$ in other words I want to sum $e^{-\frac{x^2+y^2}{2}}$ on all the integer coordinate pairs of an infinite grid. Sorry for bothering, and I…
user307181
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How can we prove the sum of squares/cubes/etc is always a polynomial of appropriate degree?

What elementary proofs are there that $$\sum_{k=1}^{n}k^m$$ is always a degree $m+1$ polynomial? It is well known that $$\sum_{k=1}^{n}k^2=\frac{n(n+1)(2n+1)}{6}$$ I'm interested in sums like this for $m\in\mathbb{N}$, $$\sum_{k=1}^{n}k^m$$ If…
GPhys
  • 1,539
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OIES formula for summation not working

I have the following summation: $$F(k)=\sum\limits_{n=1}^k\sum\limits_{d|n}\gcd\left({d},{\frac{n}{d}}\right)$$ At this OEIS link (http://oeis.org/A055155), this exact summation is found. (Credits to Lucian for pointing this out on my previous post…
Thev
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