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
2
votes
1 answer

Does there exist a closed form?

I wish to find a closed form for $\sum_{i=1}^n\frac{1}{i}$. does it exist? If so, what is it? I cannot arrive at one using any methods I am aware of.
bob the pie
  • 489
2
votes
4 answers

How do I read this triple summation? $\sum_{1\leq i < j < k \leq 4}a_{ijk}$

How do I read this triple summation? $$\sum_{1\leq i < j < k \leq 4}a_{ijk}$$ The exercise asks me to express it as three sumations and to expand them in the following way: 1) Summing first on $k$, then on $j$ and last on $i$. 2) Summing first on…
YoTengoUnLCD
  • 13,384
2
votes
1 answer

Semantic question about summations

This is a simple question about the syntax of summations. Is the following true: $\lambda_1\sum_\limits{j=1}^{\infty} f_1(E_j)+…
Jimmy2Goons
  • 456
2
votes
2 answers

Value of the sum: $\binom{19}{0} - 1/2\binom{19}{1} +1/3\binom{19}{2} - 1/4\binom{19}{3} ...... -1/20\binom{19}{19}$?

How do I find the value of this sum? I tried taking out the equal binomial coefficients as factors but this didn't really simplify anything. I am stumped.
John Doe
  • 1,080
2
votes
3 answers

How to evaluate a summation when index of inner sum cannot be equal to outer sum

I am looking at a summation that resembles the summation below: $$\sum_{i=1}^n\sum_{\matrix{j=1\\j \not= i}}^n 1$$ What is the best way to think about this summation and thus get the result? Can the sum be broken apart into a more intuitive form?
Damien
  • 75
2
votes
1 answer

Sum of Arithmetic series

Sum of consecutive values can be found easily. But I can't figure it out how to find the closed form of the following arithmetic series? Can anybody explain it elaborately? $ S = (1) + (1+2) + (1+2+3) + (1+2+3+4) + \dots + (1+2+3+\dots+n) $. Thanks…
Shadekur Rahman
  • 45
2
votes
2 answers

Why Does $ \sum\limits_{k=0}^n \begin{pmatrix} n+1 \\ k+1 \end{pmatrix} p^{k+1} (1-p)^{n-k} $ sum to $ (1-(1-p)^{n+1}) $?

I was browsing around when I found this question: Find the expected value of $\frac{1}{X+1}$ where $X$ is binomial. I understood the solution until I hit this portion where $ \sum\limits_{k=0}^n \begin{pmatrix} n+1 \\ k+1 \end{pmatrix} p^{k+1}…
crentagon
  • 23
2
votes
1 answer

Show that there is exactly one positive integer n for which $\sum_{r=1}^n r^3+\sum_{r=1}^n r = 8 \sum_{r=1}^n r^2$

Can someone show me the working for this? Thanks AQA A Level Mathematics Further Pure 1 January 2010 Question 8(b) http://filestore.aqa.org.uk/subjects/AQA-MFP1-W-QP-JAN10.PDF And if anyone could explain to me how to type equations properly instead…
Thomas Winkworth
  • 641
2
votes
2 answers

How to evaluate the sum $\sum_{j = i+1}^{n-1} j $?

How would I go about solving this summation? $$\sum\limits_{j = i+1}^{n-1} j $$ I'm trying to figure out how to solve this summation using the fact that $\sum\limits_{i = 1}^{k} i= \frac{k(k+1)}{2}$. The answer I get with my Ti-89 is…
user210375
  • 21
2
votes
2 answers

Summation of an infinite series

The sum is as follows: $$ \sum_{n=1}^{\infty} n \left ( \frac{1}{6}\right ) \left ( \frac{5}{6} \right )^{n-1}\\ $$ This is how I started: $$ = \frac{1}{6}\sum_{n=1}^{\infty} n \left ( \frac{5}{6} \right )^{n-1} \\ = \frac{1}{5}\sum_{n=1}^{\infty}…
user41235
  • 734
  • 1
  • 6
  • 13
2
votes
3 answers

Proof By Induction of a Sum

Can someone look at my proof. I am supposed to prove by induction. The question is to prove the following: $$\sum _{i=0}^{n}{i} =\frac { n\left( n+1 \right) }{ 2 } .$$ If $n=1$ Then $$1=\sum_{i=1}^1{i}=\frac{1(1+1)}{2}=1$$ Now assume $n=k$. …
Alexis Dailey
  • 99
1
vote
2 answers

Is there any way to simplify following summation?

Is there any way to simplify following summation? $$\sum_{k=1}^n \frac{1}{k^2(k+1)^2}$$
Human Learning
  • 1,033
1
vote
3 answers

Summation Proof

I'm getting stuck halfway through this: Show that $$\sum_{i=1}^n (y_i - \bar y_s)^2 = \sum_{i=1}^n (y_i)^2 - n\bar y_s^2$$ My skills with manipulating sums are quite rusty. I multiply the left side and distribute the sum to each part. I can see that…
Nick
  • 13
1
vote
2 answers

Show that $\sum_{k=1}^{p-1}f(k)=\sum_{k=1}^{p-1}f(qk)$

I would appreciate if somebody could help me with the following problem Q: Show taht $$\sum_{k=1}^{p-1}f(k)=\sum_{k=1}^{p-1}f(qk)$$ where $\gcd(p,q)=1, f(p+x)=f(x)$
Young
  • 5,492
1
vote
0 answers

How do I convert this equation from Iverson brackets to make use of the Heaviside function?

I have the equation $\sum_{i=0}^{\infty} 2^{i}[0 \leq x - 2^{i}][x - 2^{i + 1} < 0]$ and I would like to convert the Iverson brackets to the Heaviside function. I've read this post but I'm still a bit confused as to how this would work inside the…
sdasdadas
  • 177
1 2 3
…
99 100