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
1
vote
6 answers

How can I solve this summation?

I've been trying to solve this problem for quite some time now and I can't think of how to reduce the inner summation to a smaller problem. Usually when I have variables in the upper and lower bound of the summation, I just do $$(upper bound -…
ninesalt
  • 189
1
vote
2 answers

How can I find a closed form for this partial sum. $\sum_{n=1}^{k}\frac{n^3}{3^n}$

I am looking for a way to derive the partial sum formula for $ \large \sum_{n=1}^{k}\frac{n^3}{3^n}$. I notice that the wolframalpha website evaluates it as $ \frac 1 8 \frac{-4 k^3-18 k^2-36 k+33 (3^k-1) }{3^k} $ but there is no indication how…
john
  • 860
1
vote
1 answer

How to evaluate $\sum_{n=-N}^{N} \cos^2 \left(\frac{\pi}{4}n\right)$

Ok I know that the period of this function is $T=8$. I know this can help me in some way, because this means that $\cos^2 \left( \frac \pi 4 \right) + \cos^2\left(\frac \pi 2 \right) + \cdots + \cos^2(2\pi)=4$ repeats an amount of times I am not…
davidaap
  • 708
1
vote
2 answers

Is $\sum_{n=1}^\infty (-1)^n$ zero or undefined?

$$\sum_{n=1}^\infty (-1)^n$$ Is this mathematical expression zero or undefined? I think it looks like zero but i can't explain the reason mathematically. In addition, $\infty - \infty$ is undefined afaik?
mnrl
  • 113
1
vote
3 answers

$\sum_{i=1}^n a_i=\sum_{i=0}^{n-1} a_{i+1} $

how to prove that: $$\sum_{i=1}^n a_i=\sum_{i=0}^{n-1} a_{i+1} $$ where $a_i$ is any statement that dependent on $i$. I was trying to prove the binomial theorem by induction,So I needed this property. However,i remember that in Spivak's calculus…
someone
  • 9
1
vote
2 answers

Skip a number in a summation

$$\sum_{n=1}^{10} n^2$$ Returns: $1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100$ and I would like it to return: $1 + 9 + 25 + 49 + 81 + 121 + 169 + 225 + 289 + 361$ How will I go about this and, more importantly, how does it work?
HarshWombat
  • 13
1
vote
6 answers

Sum of first $n$ triangle numbers, without induction

Background I wish to calculate $$ S= \sum_{i = 1}^{n}\frac{k(k+1)}{2}$$ I know what the answer is going to be, since this is essentially the sum of the first $n$ triangle numbers. I.e. $S = (1) + (1+2) + (1+2+3) + \cdots + (1+2+3+\cdots+n)$ All…
Alec
  • 4,094
1
vote
1 answer

Sum of even digits

Let $E(n)$ denote the sum of the even digits of $n$.For example,$E(1243)=2+4=6$. What is the value of $E(1)+E(2)+E(3)+....+E(100)$? I got the answer after literally adding like $2+4+6+8+2...+6+8=400$ (for all $100$ numbers) Is there an easy tricky…
Ushosee
  • 27
1
vote
4 answers

Proofs involving summations

I am working on the following proof. Prove that $\displaystyle \sum_{k=1}^{n} \frac{2k^4+1}{k+2} \geq \frac{n(n+1)(2n+1)}{6}$. My first question is if the above is equivalent to saying "Prove that:$\displaystyle \sum_{k=1}^{n} \frac{2k^4+1}{k+2}…
Joe
  • 955
1
vote
3 answers

Specific double summation problems

How should I evaluate double sum of a number? Example $$\sum_{i=0}^n\sum_{k=i+1}^n 4 $$ And how should I evaluate double sum of $$\sum_{i=0}^n\sum_{k=i+1}^n i^2 $$ Note:- I am completely new at these. I am good at single summations though. So if…
mathnoob123
  • 1,373
1
vote
1 answer

How to solve equations that consist of summations in both sides

I have a function $f$ defined as follows, $$f(n) = \sum_{1 \leq i \leq n} i(i-1)$$ but I want to find $n$ and $n'$ for which the following holds, $$\frac{f(n)}{n(n-1)} = \frac{3f(n'/3)+6f(2n'/3)}{n'(n'-1)}$$ I tried a few random numbers, but I…
orezvani
  • 394
1
vote
1 answer

How to get the last line?

I'm supposed to find the following equality: $\sum_{r=1}^{s-1}2^{2r-1} -\sum_{r=1}^{s-1}2^{r-1} +(n-2^{s-1}+1)(2^s-1)$ $=\frac{2}{3}(4^{s-1}-1)-(2^{s-1}-1)+(2^s-1)n-2^{2s-1}+3*2^{s-1}-1$ I understand that:$\sum_{r=1}^{s-1}2^{r-1}=(2^{s-1}-1)$ and…
astudentofmaths
  • 696
1
vote
1 answer

Isolate x in the equation

I'm trying to isolate $x$ from the next equation. Some ideas? \begin{equation} k_{1} = k_{2}\sum_{i=1}^{x}\binom{x-1}{i-1}\frac{1}{i^{k_{3}}}, \end{equation} where $k_{1}$, $k_{2}$ and $k_{3}$ are constants. If not possible to do it exactly, an…
Henry
  • 33
1
vote
2 answers

Question about simplification in summation

I am reading a book where the following example is shown. $$= \sum_{0\le n-j\le n} (a+b(n-j)) $$ $$= \sum_{0\le j\le n} (a+bn-bj) $$ Why is n-j being simplified to j? I don't understand why this is possible? To specify my question, what rule of…
Thomas Christopher Davies
  • 145
1
vote
4 answers

Summation $ \sum_{k=-\infty}^n a^k$

Is there any formula that directly gives the result of this summation: $$ \sum_{k=-\infty}^n a^k$$ ?
Uygar Uçar
  • 33
1 2 3
…
99 100