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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Is it possible to express $\frac{1}{n}+\frac{1}{1+n}+\dots+\frac{1}{2n-1}$ as a sum

I have the following question: Let $f:[1,2]\to\mathbb R$ defined by $f(x)=1/x$. Prove $∫_1^2 \frac{1}{x} dx=\log⁡ 2$. In the sample answers i have given the dissection $D_n= [1,r,r^2,...,r^n] $ with $r=2^{1/n}$. To get more experince i am trying…
leahforbes
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Sigma sign issue

In the answer key to one of the problems, the following step takes place $f(x)=\sum_{k=1}^n\left(a_kx^2-2x+\frac{1}{a_k}\right)=(a_1+a_2+...+a_n)x^2-2nx+\left(\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}\right)$ Why does $\sum_{k=1}^n2x=2nx$ not…
Pen and Paper
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How can I make this summation valid?

I've been told that the above summation is wrong because the part to the right of the summation has to be some function of K. The expression I want to show is if n is 100 (people) add together z (their age) and divide by q. If I can't use z is…
Pattle
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Intuition on Changing Order of Summations

$$ \sum_{i=1}^{\infty}\left[\sum_{j=i}^{\infty}f(i,j)\right]=\sum_{j=1}^{\infty}\left[\sum_{i=1}^{j}f(i,j)\right] $$ I have trouble understanding the change of order of this summation. I know that $$ 1≤i≤j≤\infty $$ however, I can't seem to make…
wd violet
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How would I convert a recursive geometric sequence summation to a closed formula?

How would I convert the below geometric (I assume based on the terms) recursive sequence summation to a closed formula? $$a_1 = 1,\ \quad a_k = \sum_{i=1}^{k-1} a_i \ \quad for\ k \geqq 2$$ I've tried: $$a_k = 2\frac{k-1}{k}$$ $$a_k =…
Jess
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How do I prove this is true? $A(j+x+x^2+....+x^n)= A(j-x{^{n+1}})(j-x){^{-1}} $

How do I show that the following is true? $$ A(j+x+x^2+....+x^n)= A(j-x{^{n+1}})(j-x){^{-1}} $$ I tried dividing both sides by $A$ and then multiplying by $(j-x)$, but I'm not sure how to proceed from there because the $j$ is throwing me off.…
BirbCS
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Strange Summation Identity

I was looking at the Summation wikipedia(https://en.m.wikipedia.org/wiki/Summation) and found this rather strange identity: $$ \sum_{k\leq j\leq i\leq n} a_{i,j}=\sum_{i=k}^n\sum_{j=k}^i a_{i,j}=\sum_{j=k}^n\sum_{i=j}^n…
Ayush Anand
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Interchange the order of summation

if I have $$ \sum_{n=0}^{\infty}\sum_{p=0}^{n} (\cos)^{n-p} (i\sin)^{p} x^n $$ Now substituting $x^n= x^{n-p}x^{p}$ and using the transformation identity $$\sum_{j=1}^{\infty}\sum_{i=1}^{j}f(i,j)=\sum_{i=1}^{\infty}\sum_{j=i}^{\infty}f(i,j)$$ Here…
AEIOU
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$\sum_{i,j=1}^N \frac{1}{(i^2 + j^2)}$ (i..e. $1/r^2$ over the square lattice $i,j\in\{1,2,\ldots,N\}$)

Does anyone have any insight as to how I might find a closed-form expression for $\sum_{i=1}^N \sum_{j=1}^N\frac{1}{(i^2 + j^2)}$ ? It feels like it should be straightforward because it's easy to do as an integral in polar coordinates if you include…
jms547
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What will be the value of floor function of $\lim\limits_{N\to\infty}\left\lfloor\sum\limits_{r=1}^N\frac{1}{2^r}\right\rfloor$

What would be the value of floor function of $\lim\limits_{N\to\infty}\left\lfloor\sum\limits_{r=1}^N\frac{1}{2^r}\right\rfloor$ would it be $1$ or would it be $0$ ? The formula I use for this is that of infinite summation series that is…
marks_404
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Let $H=\{2,3,4,\dots,n+1\}$. Show that $\sum_{\emptyset \neq S\subset H}\prod_{i\in S}\frac{1}{i}=n/2$.

Let $H:=\{2,3,4,\dots,n+1\}$. Show that $$\sum_{\emptyset \neq S\subset H}\prod_{i\in S}\frac{1}{i}=n/2.$$ For example, with $n=3$, we have $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{2\cdot 3}+\frac{1}{2\cdot 4}+\frac{1}{3\cdot…
tmaj
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The sum of the terms in the nth group

Let the natural numbers be divided into the following groups: $ ${1}$,${2,3,4}$,${5,6,7,8,9}$.....$ What is the sum of the terms in the $n$th group? I know that the number of terms in nth group will be $2n-1$. But, I am not able to get a general…
Adienl
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Are these two equations equivalent

Can I say that this equation is equivalent to $$\frac{1}{S}\frac{1}{U} \sum_{p=1}^{S} \sum_{u=1}^{U} PL . SF (|\alpha_{0,u,p}|^2+\sum_{n=1}^N\sum_{m=1}^M|\alpha_{n,m,u,p}|^2 )$$ Thank you.
Tyrone
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How to isolate variable in a sum when result is known?

I'm building a little tool to help configuring a Martingale-style laddered trading strategy. Assuming the following variables: P = Final Position Size S = Number of ladder steps I = Size Increment F = First position size $$P=\sum_{i=0}^{S-1}…
thisisbbc
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Showing $\sum_{i=1}^n\sum_{j=1}^n\frac1{ij}=\sum_{i=1}^n(\sum_{j=1}^i+\sum_{j=i}^n)\frac{1}{ij}-\sum_{i=1}^n\frac1{i^2}$

I was reading Almost Impossible Integrals by Cornel, where I encountered this manipulation $$\sum_{i=1}^n\sum_{j=1}^n \dfrac{1}{ij}= \sum_{i=1}^n \left(\sum_{j=1}^i+\sum_{j=i}^n\right)\dfrac{1}{ij}-\sum_{i=1}^n\dfrac{1}{i^2}$$ I am new to such…
Laxmi Narayan Bhandari
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