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

Compute $\sum\limits_{a=1}^{\infty} \sum\limits_{b=1}^{\infty} \sum\limits_{c=1}^{\infty}\frac{ab(3a+c)}{4^{a+b+c}((a+b)(b+c)(a+c)}$

Compute the following sum: $$ \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \sum_{c=1}^{\infty}\frac{ab(3a+c)}{4^{a+b+c}((a+b)(b+c)(a+c)}\ $$ I have no head or tail of how to even start this since the summation is interlinked.
Ashwin Krishnan
  • 111
2
votes
3 answers

Express $S =\frac{1}{1\cdot2}+\frac{1}{3\cdot4}+\ldots+\frac{1}{(n-2)\cdot(n-1)}+\frac{1}{n\cdot(n+1)}$ in terms of $n$.

Express $S =\frac{1}{1\cdot2}+\frac{1}{3\cdot4}+\ldots+\frac{1}{(n-2)\cdot(n-1)}+\frac{1}{n\cdot(n+1)}$ in terms of $n$. Here's what I have done so far: $$\begin{align*} S…
Huye
  • 23
2
votes
2 answers

Summation simpilification process

Here is my summation: $$\sum_{n=0}^{N/2-1}\frac1{25+nB}=A$$ where $0.01\le A\le2$, $10\le N\le2000$ and I need to find the $B$ for different values of $N$ I calculate this summation online (check here) But the Digamma function makes the output…
John Jin
  • 145
2
votes
2 answers

Digit Sums: A Math Project

I'm doing a Math project on Digits Sums, and I don't really want to try to figure out all the 4-digit numbers that add up to 2, 3, 4, 5, and 6. That's partly because it would be hard, and also, because I don't want to get anything wrong, or it would…
Confuddled
  • 21
2
votes
1 answer

summation of binomial series

Can someone pls provide the approach/solution to this problem $$\sum_{r=1}^n \begin{pmatrix}n\\r\end{pmatrix}\sin(rx) \cos((n-r)x)$$ I have tried to simplify the sine and cosine terms and then arrived at this part but am unable to solve any…
ErrorEliminator
  • 45
2
votes
3 answers

question about summation?

Are there any general rules to find $???\leqslant \sum_{n=t}^{m}f(n)\leqslant ???$ when $m$ and $t$ $\in $ R
mnsh
  • 5,875
2
votes
1 answer

Writing this concatenating decimal as a series/fraction

I am trying to work out a way to write the number $0.123456789101112...$ as a series, I thought at first doing something like: $$\sum_{n=1}^N\frac{n}{10^n}$$ but when we get into the double digits this doesn't work, so how should I do this? Also is…
Henry Lee
  • 12,215
2
votes
2 answers

$x$ as the shortest alternating sum of $1 \ldots n$

If I have an positive integer $x \in \mathbb{N}$ and I have $Z = \sum_{i = 0}^{n}{i}$ such that $Z \geq x$ and $Z - x \equiv 0 \bmod 2$ and $n$ is the smallest such integer it is possible to create and alternating sum from $1 \ldots n$ such that it…
Daniel Papp
  • 23
2
votes
1 answer

Sum of a constant over a domain

After some research, I cannot find a solution to this very basic question: What is the sum of a constant over a domain (of a variable X): $$\sum_{X \in \{0,1\}} 1 = ??$$ I would say that the result is $2$ but I am not sure. Thanks by…
ailauli69
  • 143
2
votes
1 answer

Find a starting value such that the value of the function is always positive

Suppose I have a variable number of integers, each of which can be of any value from negative to positive infinity, and not sorted in any particular order. For example: -1, -1, -1, -2, 9 The "function" is evaluated by picking a starting point, and…
404 Name Not Found
  • 123
2
votes
1 answer

Sequence of integers such that the sum of two disjoint subsequences is different

I would like to know if it is possible to generate a sequence of integers $S = x_0, ..., x_n$ such that, if $Q$ and $P$ are two disjoint subsets of $S$ (even of different cardinality), and $C = c_0, ..., c_{|P|}$ is a sequence of integers greater…
weghiuop
  • 43
2
votes
1 answer

Why does the following sum substitution work?

$$ =\sum_{i=1}^{n} \frac{(n-i)(n-i+1)}{2} $$$$ =\sum_{i^{\prime}=0}^{n-1} \frac{i^{\prime}\left(i^{\prime}+1\right)}{2} $$ I see that $i' = n - i$, but shouldn't then the bounds be different? Like for the lower bound $n - 1$ as you substitute in $1$…
Hilberto1
  • 1,046
2
votes
4 answers

What is this sum $\sum_{i=0}^{n-1}\sum_{j=0}^{i}x^i$?

How to find this sum ? $$\sum_{i=0}^{n-1}\sum_{j=0}^{i}x^i$$ without knowing this sum: $$\sum_{i=1}^{n}ix^{i-1} = \frac{nx^{n+1}-(n+1)x^n+1}{(x-1)^2}$$
Jotadiolyne Dicci
  • 374
2
votes
2 answers

Simplification for a double summation where the upper limit of the inner index depends on the value of the outer index?

$$\sum_{x=1}^{N} \sum_{y=1}^{M(x)} (1 + a\cdot f\left(x\right))(1 + b \cdot f\left(y\right)) \tag{1}$$ where $N$, $a$, and $b$ are integer constants. $M$ is also an integer but changes for every value of x, which makes the index of the second…
Hitanshu Sachania
  • 141
2
votes
2 answers

Summation of a Summation of two variables

I am taking a physical chemistry course and some of our practice problems include summations of summations, which I've never dealt with before. I have the answer, but I don't understand where it came from - why aren't there any terms with $ i $ or $…
jmm5180
  • 21
  • 3
1 2 3
…
99 100