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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Calculating Riemann sum

A cannonball is shot into the air. Its velocity is given as a function $f(t) m/s$, where $t$ measured in seconds since $1:00$ PM. We know that $f (t)$ takes the following values: t 0 7.5 15 22.5 30 37.5 45 52.5 60 f(t) 10.0 6.46 5.00 3.88 2.93 2.09…
mathnoob123
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How would I find a closed form expression of this sum?

Earlier I was looking to find a closed form expression of the sum: $$\sum_{h=1}^{k-1}\left({2^{h}}{3^{k-h-1}}\right)$$ Wolfram Alpha tells me that this is equivalent to: $$2\times{3^{k-1}}-2^k$$ However it does not explain the method of how it got…
ketchupcoke
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Prove the formula for the sum of the first N odd cubes.

Using this formula: $1^3+2^3+⋯+n^3=[\frac{n(n+1)}{2}]^2$ Prove: $1^3+3^3+⋯+(2n+1)^3=(n+1)^2(2n^2+4n+1)$ I had a hard time trying to prove this, so I'll be glad if someone could help me. This is a exercise from the book "What is Mathematics?"…
user455937
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Is the subtraction of two summations with sequential stopping points equal to the typical element?

If I have two summations, $\sum_{k=1}^{n} k^2$ and $\sum_{k=1}^{n+1} k^2$ and I subtract the summation with the larger stopping point from the one with the smaller stopping point, like so $\sum_{k=1}^{n+1} k^2 - \sum_{k=1}^{n} k^2$ will the result…
McFizz
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Name and proof of property of summation - Change sign index

What is the name of the following property of summation (t is the time)? $$\sum_{i=a}^{b}x_{i}(t)=\sum_{i=-b}^{-a}x_{-i}(t)$$ Could you show me a proof of it please? Thank you. PS: if possible show me webpages of the above property please.
Gennaro Arguzzi
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Two different results for the summation $\sum_{n=1}^{\infty}n^2 p^{n-1}$?

I'm trying to calculate the the following equation: $$ p(1-p)\frac{d}{dp}(p\frac{d}{dp}\sum_{k=0}^\infty p^k) $$ I've expanded the summation in the following steps: $$p(1-p) \frac{d}{dp}(p \frac{d}{dp}[1 + p + p^2 +p^3 +....])$$ $$p(1-p)…
MarksCode
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Having confusion in this how can I solve

An ellipse is drawn with major axis and minor axis of lengths 10 and 8 respectively. Using one focus as the centre,a circle is drawn that is tangent to the ellipse,with no part being outside the ellipse. Then the radius of the circle is: Ans :…
Aakash Mishra
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Is there a way to simplify this summation further?

I've been doing a bit of summation work, and I got to$$\sum\limits_{k=1}^{4n+1}\frac 1{n+k}-\sum\limits_{k=1}^n\frac 1{5k-3}-\sum\limits_{k=1}^{n}\frac 1{5k-2}+\frac 35\sum\limits_{k=1}^n\frac 1k\tag1$$And I'm wondering if there is a way to further…
Crescendo
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How are this two summations are equal [Basic summations]

Can anyone explain to me how the following two summations are equal? $=\sum_{j=i+1}^n j$ $=\sum_{j=1}^n j - \sum_{j=1}^i j$
jason
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Calculating finite sum using only pen and paper

I know this is something that will pop up for my math exam tomorrow. When searching for this the only thing I could find was this: Calculate $\sum\limits_{n=1}^\infty (n-1)/10^n$ using pen and paper, but this is for infinite sums. So I'd like to…
RandomDude
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Sum of integer sequence with occasional sign change $\displaystyle\sum_{k=1}^{2018}(-1)^{^{\binom {k+2}3}}k$

This question was designed based on this other question here. Evaluate $$\sum_{k=1}^{2018}(-1)^{^{\binom {k+2}3}}k$$ and explain why it would be a perfect square. Generalize.
Hypergeometricx
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Why is this expansion true?

Why are this two sums equal? $$ \sum_{i=1}^n i2^i = \sum_{j=1}^i\sum_{i=j}^n 2^i$$ I'm supposed to prove that both sums are the same.
Arie Wortsman
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What is the meaning of the symbol H in the formula?

It's a formula I've got after finding sum. But I don't understand what H means
Kyrylo Niedieliev
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Evaluate: $ \sum_{i}^{n+1} n !$

$ \sum_{i}^{n+1} n !$ This sum is suppose to equal $(n+1)!$. For some reason I don't get why. Can anyone explain? Taken directly from http://www.inchmeal.io/2016/01/15/how-to-prove-it-ch-7-sec-1.html q22(a) last part.
helios123
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Summation of odd squares up to n

So I am trying to figure out a summation of all odd squares up to n. I.E if n = 9, then the output should be 10 (the odd squares <= 9 are 1 and 9, therefore 1+9 = 10). Can anyone help me out? I can't wrap my head around this one. I did one fore the…
Jacob R
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