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.

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How to calculate the sum : $\sum_{k=1}^{n}\frac{1^2\space5^2\space...\space(4k-3)^2}{3^2\space7^2\space...\space(4k-1)^2}$?

I need to calculate the sum : $\sum_{k=1}^{n}\frac{1^2\space5^2\space...\space(4k-3)^2}{3^2\space7^2\space...\space(4k-1)^2}$ I am thinking of writing it as a difference of sums so when I do the difference some terms will reduce and we will get the…
Ghost
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lframConvergence of $\sum_{n=2}^\infty \frac1{\log(n!)}$

How do I show that this sum diverges/converges? $$\sum_{n=2}^\infty \frac1{\log(n!)}$$ I want to use the comparison test, but I do not know how to approach. Also, Wolfram says this diverges by the comparison test, but Mathematica gives me a…
John Glenn
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Proving simplification of summation

I'm trying to prove that $\frac{1}{1+z^2}=1-z^2+z^4-z^6+...$ for $|z|<1$. The only thing that I can think of is that $\sum_{n=0}^\infty(-1)^nz^{2n}=1-z^2+z^4-z^6+...$, but I'm rather lost. Can anyone give me a hint please?
AdamK
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How can I calculate $\sum_{i=1}^j \sum_{j=1}^4 \left(2ij+1\right) $ by doing the second summation after the first one?

Consider $$\sum_{i=1}^j \sum_{j=1}^4 \left(2ij+1\right) $$ How can I calculate the second summation after the first one?
Mohd Alfendi
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How to find closed form of summation

How do you find the closed form of this summation? $$\sum_{i=0}^{\log_4 n-1} i^2 $$ I know the following: $$\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$$ How can I use this to find the closed form of my summation? Thanks
kelp99
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Can you algebraically invert an infinite sum?

If you have two different kinds of series representations like $$ \sum_{n=1}^{ \infty}f(n)=\sum_{k=1}^{ \infty}g(k),$$ does it follow that $f(n)=g(k)$?
user19788
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Basic Maths...can’t get my head around this.

This might be very basic but... Take any number for example 2356 now 2+3+5+6=16 and 1+6=7 or 236+5=241 and 2+4+1=7 and also 21+4=25 again 2+5=7 or 652+3=655 which digits sum is 7 again and then 65+5=70 again 7 or 56+5=61 again 7 And so on make any…
jasb1327
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variable substitution in multiple summation

I have a question about variable substitution in summation and I don't know the answer. Didn't find the answer by searching, thought of asking it here. Assume a polynomial matrix $P(\alpha)$ is written as: $P(\alpha) = \sum_{k=0}^{g} P_k…
Keivan
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Special cases where rearranging the order of a summation preserves value / divergence

This question shows a special case of an infinite double sum where rearranging the order of summation preserves the value of the expression* $$ \sum_{i=1}^{\infty}\sum_{j=(i)}^{\infty}f(i,j)=\sum_{j=1}^{\infty}\sum_{i=1}^{j}f(i,j) $$ Or, written…
Greg Nisbet
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Finding $\sum_{k=1}^n k \text{ } 2^k$

I need to calculate this sum: $$\sum_{k=1}^n k \text{ } 2^k$$ I tried to meddle with double sums using $$\left ( \sum_{k=1}^n k\right )\left( \sum_{j=1}^n 2^j \right)$$ but It doesn't seem to be the most fitting approach.
fuzzypixelz
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How to Solve Summation Notation

I am having trouble with this equation in a class I am taking and I am trying to understand it: $$\sum_{i=47}^{i=136} M_i$$ We have to solve for the problem below but the hint our professor gave us was subtraction. I am confused because it is my…
Schmit
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Summation of simple series $\sum_{r=5}^n (2r + 4r^2) $

I am trying to teach myself, but I am confused on one question. It says "for the following summation, give an equivalent equation without the summation: $$\sum_{r=5}^n (2r + 4r^2) $$ where $i$ takes values from $5$ to $n$.
jessica.m
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A question about $\sum_i^n (A+BX_i)$

$$\sum_{i=1}^n (A+BX_i)= nA +B\sum_{i=1}^n X_i$$ Not sure how to express this question, as english is not my first language. I know why the A receives an n, but my question is, why does B receive none? B and $X_i$ are separate so shouldn't B…
Crimson Crusader
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Converting a summation function into a continous function.

Consider functions like this: $$\sum_{n=1}^xn=\frac{x\left(x+1\right)}{2}$$ For any polynomial the summation can be turned into a continuous one. https://en.wikipedia.org/wiki/Faulhaber%27s_formula Also consider: $$\sum_{n=0}^x\sin…
Nimish
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Rearrange the index of a summation to get an infinite sum

I'd like to know the basis for the following transformation: $$\sum_{i,j:i+j=k}a_ib_j \quad k=0, \dots n+m$$ Let $j = k-i$ then: $$\sum_{i=-\infty}^{\infty}a_ib_{k-i} \quad k=0, \dots n+m$$ I understand that the $j$ index is eliminated by the…
Dianne
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