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
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How to simplify this equation: $1\cdot N + 2(N-1) + 3(N -2) + \cdots + i(N - i + 1) + \dots + N \cdot1$?

$$1 (N) + 2(N-1) + 3(N -2) + \cdots + i(N - i + 1) + \cdots + N (1) $$ I need to write it in simplest form? here 1(N) means 1 multiply by N
Grijesh Chauhan
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How to convert this into a single closed form?

I have a loop that iterates a variable $i$ from 0 to some boundary $b$. Let $p=4i+4$. Each iteration of the loop I add $(p^3+5p)/3+2$ to a sum variable $s$ which is initially set to 0. Question: Instead of this loop, how can I convert this all into…
MyNameIsKhan
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how is sum of sines related to digamma?

I was studying a technique in Gradshyten and Ryzhik's Part 12 "Some Log Integrals" on how to integrate something like $\displaystyle \int_{0}^{1}\frac{\ln(x)}{x^{2}-x+1}dx$ and came across a relation I am wondering about. The paper uses the roots…
Cody
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Evaluate the sum? problem..

I need to evaluate the sum: can someone help? The series is as follows: $$ \frac 14 - \frac {1}{2(4)^2} + \frac{1}{3(4)^3} - \cdots + \frac{(-1)^{(x+1)}}{x(4)^x} $$
mc1
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Hint about a identity

Who has hint how to prove: $\sum_{n=0}^N \sum_{k=1}^N g_k y_n = \sum_{n=1}^N \sum_{k=1}^n g_k y_{n-k}+\sum_{n=1}^N\sum_{k=1}^n g_n y_{N-k+1}$ Thank in advance!
TheStudent
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In a double summation, can the range of the outer sum depend on the index variable of the inner sum?

Given this summation:$$\sum_{1\le y\le x\le 3}2x+y=\sum_{x=1}^3\sum_{y=1}^x2x+y=\sum_{y=1}^3\sum_{x=y}^32x+y$$Am I allowed to write the summation as:$$\sum_{y=1}^x\sum_{x=1}^32x+y\space\space?$$ Basically, my question is: In a double summation, can…
A Slow Learner
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Modification to geometric infinite summation

I've been exploring the collatz conjecture ($3n+1$ problem) as an interest and came up with the following problem. Would greatly appreciate if help could be found! Typically, for geometric summations to infinity, typical textbooks deal with…
brorion
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Find the original function in a summation

I have the sum $$S=\frac 12 \sin x$$ and need to find the original function to this summation equation: $$\sum_{n=1}^\infty f(x)=S$$ I am using the math software Maple, but can't solve this equation there. Is it possible to find the (or an) original…
Steeven
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how to interpret this double summation, the first from i=1 and the second from j does not equal i

I'm not sure what the meaning of the double sum is in this formula.
user342836
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How to prove that $2^0 + 2^1 + 2^2 + 2^3 + \cdots + 2^n = 2^{n+1} - 1$

When I'm reading Computer Systems: A Programmer's Perspective, I met the sum of binary numbers and failed to prove it: $$ 2^0 + 2^1 + 2^2 + 2^3 + \cdots + 2^n = 2^{n+1} - 1 $$ This might be preliminary knowledge, I'm not good at mathematics, any…
Nick
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Summations with backwards limits

I have a sum defined as $ \sum_{\mu=0}^{k-1} f(\mu) $ , what does it mean in the case that $k=0$? Then I have a sum running from $0$ to $-1$ which I suppose doesn't make sense? Wolfram alpha gives $0$ no matter what I put in for $f(\mu)$, why is…
calvingentry
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Concrete Mathematics: Manipulation of Sums clarification

For reference this on on page 31 of the first edition of Concrete Mathematics by Graham, Knuth, Patashnik. Let K be any finite set of integers. The commutative rule for sums is stated as follows: $$\sum_{k\in K}a_k = \sum_{p(k) \in K}a_{p(k)}…
Mike. E
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how to compute this summation /composite function

Let $f(x)$ be the set of all even numbers smaller or equal than $x$. For example, $f(10)=\{0,2,4,6,8,10\}$. Let $g(x)=2x$. Consider the double sum: $$\sum_{i=1}^4 \sum_{j\in(g \circ f)(i)} j$$ I know that $(g \circ f)(i)=g(f(i))$ Where should the…
user1798157
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Please help me to calculate the sum $t \sum_{s=0}^{t} \frac{ (-1)^{t-s} s^{t-s-1} e^s }{(t - s)!}$

I need to prove that the following sum $$t \sum_{s=0}^{t} \frac{ (-1)^{t-s} s^{t-s-1} e^s }{(t - s)!}$$ tends to 2, when t goes to infinity. Here $e=2.718281828459...$
Vahagn Poghosyan
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How to get a sum of unique numbers always unique? How to pick such numbers?

I have a requirement where I have to label each node in a graph with a number. So that when I reach a leaf node traversing a path in that graph I can use the sum of all the node labels in that path to identify the path. I know powers of two work.…
Ruby9191
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