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
0 answers

Visual/intuitive proof of why $\sum k^3 = (\sum k)^2$, where $k$ goes from 1 to $n$?

I understand that one could prove this by first proving the analytic expressions of the sigma terms through induction, and then square the $\displaystyle\sum_{k=1}^n k$ term to show LHS = RHS. Are there any other easier to understand (preferably…
JDS
  • 713
1
vote
0 answers

How to solve this summation + Generalizing?

we know that $$\sum_{i=1}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$or $$\sum_{i=1}^n i = \frac{n(n+1)}{2}$$ and we can prove this by telescoping series $$ (i+1)^2-i^2=2i+1$$ $$(i+1)^3-i^3=3i^2+3i+1$$ But while solving some problems, I suddenly thought about…
Ivan_Raki
  • 73
1
vote
2 answers

Summation of a multiplication

How can I prove $$\sum_{k=1}^\infty kq^{k-1}=\frac{1}{(1-q)^2}$$? I know that the formula to do the summation of a number $a$ is $a(a+1)/2$. And I also know that the summation of a geometric series $ar^n$ is $a/(1-r)$. But I don't know how to solve…
User160
  • 953
1
vote
0 answers

What is the sum of 1/n^n as n goes to infinity?

What is the actual value of the sum of 1/n^n as n goes to infinity? It's about 1.29129.
lurf jurv
  • 111
  • 1
1
vote
3 answers

Summation of sequence $a_n - a_{n-1} = 2n$

$(a_1,a_2,a_3,..)$ be a sequence such that $a_1$ =2 and $a_n - a_{n-1} = 2n$ for $n \geq 2$. Then $a_1 + a_2 + .. + a_{20}$ is equal to? $a_1$ = 2 $a_2$ = 2 + 2x2 $a_3$ = 6 + 2x3 $a_4$ = 12 + 2x4 $a_5$ = 20 + 2x5 $a_n$ = $b_n$ + $g_n$ ,here…
Zero
  • 345
  • 1
  • 4
  • 9
1
vote
1 answer

A legal reordering of two sigma notations

Imagine you have two finite sets, A and V, and a function $f: A \rightarrow \wp(V)$ (in my case A is a subset of V but that does not matter). We also have a function $g(a,b)$ where a is an element of A, and b is an element of V, and a function…
RetAFVLib
  • 30
  • 6
1
vote
2 answers

iterate sum with powers

Let $R$ be a non commutative ring and $f$ is a function such that $f(ab)=f(a)f(b)+f(a)b+af(b)$ with $a, b \in R$ Calculate $f(x^n)$ My attemt: $f(x^n)=\sum_{i_1 \leq i_2 \leq ... \leq i_{n-1}\leq n-1}x^{i_1}f^{i_2}(x)......x^{i_{n-1}}f^{i_n}(x)$ but…
سعيد بلقاضي
  • 41
1
vote
1 answer

algorithm efficiency

for (int i=1;i<=n;i++){ for (int j=i;j<=n;j++){ do_something } } I need to calculate how many times the "do something" step happens. I started like so: $\sum _{i=1}^n\:\sum _{j=i}^n\:1$ I got stuck here trying to open the inner…
DarkLeader
  • 256
1
vote
3 answers

Express the following sum in terms of $n$: $\sum_{k=1}^{n} ( \frac{1}{2k-1}-\frac{1}{2k+1})$

I want to express this sum in terms of $n.$ $$\sum_{k=1}^{n} \left(\frac{1}{2k-1}-\frac{1}{2k+1}\right)$$ I've read somewhere that the sum should be equal with $\frac{2n}{2n+1}$, but I don't see how I could reach that result. Thanks for any help.
Karin Angelis
  • 29
1
vote
1 answer

Moving terms outside of summation

If i have a sum say $$C\sum_{l=1}^C\left( a+b\cdot c_l\right)$$ Is is true in general that $$C\sum_{l=1}^C\left( a+b\cdot c_l\right) = a+\left( C\sum_{l=1}^C\left(b\cdot c_l \right) \right)$$ and $$C\sum_{l=1}^C\left( a+b\cdot c_l\right) =…
user488081
1
vote
0 answers

Proof of a summation identity

Let $\vec{x}=(x_1,x_2,x_3,x_4,x_5)$ be a binary string of length $5$ i.e. $x_k\in\{\pm1\}$. Let $\vec{a}_1,\ldots,\vec{a}_5$ be some arbitrary 3D vectors (this is not essential for the problem, I can have any 5 independent symbols here instead of…
Никола Андрејић
  • 37
1
vote
1 answer

How do I evaluate $\sum_{i=0}^{n-1}2^i+1$ and $\sum_{i=2}^{n-1} \log i^{n}$?

I'm having trouble with finding the sums of the problem $\sum_{i=0}^{n-1} 2^{i+1}$ and $\sum_{i=2}^{n-1} \log i^{n}$ I've thought it over and don't know where to start with either of these problems.
Mike Johnson
  • 13
1
vote
1 answer

Get the initial value of a summation with known formula and result?

I am currently studying programming and one of the problems I had wanted me to get the initial value $n$ of the following summation which I know the result $m$ (the problem isn't exactly this, but I turned the problem into…
Spiwocoal
  • 11
1
vote
1 answer

Show that $\sum_{i=1}^{k}{k+1 \choose i}S_n(i)=(n+1)^{k+1}-n-1$?

I've been trying to answer the second problem here: And the hint is: I am having the following problem, If I expand the LHS, I get: $$\sum_{i=1}^{k} {k+1 \choose i} S_n(i)={k+1 \choose 1} S_n(1)+{k+2 \choose i} S_n(2)+ \dots + {k+1 \choose…
Red Banana
  • 23,956
  • 20
  • 91
  • 192
1
vote
1 answer

How to evaluate the closed form of the sum with factorials?

I have got the sum $$\sum_{t = 1}^{20} t \cdot \frac{(n - t)!}{n!}$$ Is it possible to get the value of this in terms of $n$ without calculator ?
NMZ
  • 175
  • 9
1 2 3
…
99 100