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.

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How to show formula for area?

Here is my formula for the area of $n$ layers of appolonian gasket(assuming no circles past the $n$th layer): $$πR^2 - \left(πR^2 - \left(\sum_0^n x_n\cdot\pi r_n^2\right)\right)$$ Here $R$ is the radius of the outer circle, $r$ is the radius of an…
Caters
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Formula for accruing costs

I am trying to work out the formula (pretty sure I leant it at some stage, but I can't for the life of me recall it), for calculating the total cost of, for example, 5 cents accruing every month, for 5 years. Eg, the first month is 5c, the second…
user394139
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Solution of this summation: $\sum_{i=1}^{10} {\frac{i}{i^4+i^2+1}}$

The summation in question: $$\sum_{i=1}^{10} {\frac{i}{i^4+i^2+1}}$$ I have been able to factorize $i^4+i^2+1$ as $(i^2+i+1)(i^2-i+1)$ but I doubt this will help. What is the solution?
Mihir Chaturvedi
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Proof $\sum\limits_{a=1}^{n} a^3 = {\left(\sum\limits_{a=1}^{n} a \right)}^2$?

Today I saw the following example and I was wondering if it's true for: ${}{}{}{}$ $$\sum_{k=1}^{n} k^{m} = \left ( \sum_{k=1}^{n} k \right )^{m-1}$$ Can someone help me out with it? Thanks!
Joshua Salazar
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Why do these two summations not equate?

Why is the below equation false? (1) $\sum_{i=0}^n i^2 = \ (1/4)\sum_{i=0}^{2n} {i}^2 $ When If we let $\ j=2i $ , and substitute $\ i=(j/2)$ into the leftmost equation above, then: (2) $\sum_{j=0}^{2n} {(j/2)}^2 $ Which gives the same sum as: $\…
JojoMojo
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Find closed form of $\sum_{j=1}^n 3^{j+1}$

What is the closed form of the following: $$\sum_{j=1}^n 3^{j+1}$$ I'm new to summations. Is it this? $$\sum_{j=1}^n 3^{j} + \sum_{j=1}^n 3$$ Then using the closed form formula: $$\frac{3^{n+1} - 1}{2} + 3n$$
ehh123
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How to Solve Summation by Hand

I am having trouble solving the following summation by hand. $$ \sum_{i=0}^{n-1} 2^i (n-i) $$ Can someone guide me in the right direction, especially for the $i \cdot 2^i$ part?
BostonMan
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How would I simplify this summation: $\sum_{i=1}^n (i + 1) - \sum_{j=1}^n j $

$$\sum_{i=1}^n (i + 1) - \sum_{j=1}^n j $$ I cant really get my head around on how to simplify these sigma notations, any help would be appreciated. Thanks
user381757
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Closed form of a sum $ \sum_{i=1}^{n}\frac{1}{((i-1)x)^2+y^2}$

Consider a sum: $$ \sum_{i=1}^{n}\frac{1}{((i-1)x)^2+y^2}$$ with $x$ and $y$ being (non-zero) constants. Is it possible to obtain a nice closed form of this expression?
user263286
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Compute the following finite sum: $\sum_{i=1}^{n}i^22^i$

Compute the following sum or give me a guidance (I try to solve it by use of logarithm but I could not ) $$\sum_{i=1}^{n}i^22^i=?$$
ebad
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Alternative summation notation

I am reading Xie and Beni's well-known paper on a fuzzy cluster validity function. In it, they use an alternative sigma notation for summation. I have seen it only infrequently and I'd love some clarification on precisely how to read it. For…
hendra
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Find the value of the summation of 3 to the power of n

I'm suppose to find the value of $$ \sum\limits_{i=5}^{100}(3)^n $$ My professor gave me the first step to this which is $$ \sum\limits_{i=1}^{100}(3)^n - \sum\limits_{i=1}^{4}(3)^n $$ and I honestly can't figure out what to do next. I did not…
Josefhu15
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Positive finite sum with restriction

Is it possible to satisfy the condition $\sum_{i=1}^N \sum_{j=1}^N\frac{q_i q_j}{a_{ij}}>0$, given that $q_i$ and $q_j$ are non-zero integers, $a_{ij}$ are positive real numbers and $\sum_{i=1}^N q_i =0$? As a special case, allow $N \rightarrow…
Nikola Poljak
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Does this sum have any meaning?

In class today I was looking at the sum $1 +\frac{1}{2}+\frac{1}{3}+\frac{1}{4}...$ and with a bit of fiddling, managed to come up with the following: $$\sum_{n=2}^\infty \left(\sum_{m=2}^\infty \frac{1}{n^m} \right) = 1$$ I managed to show this in…
Simon Goodwin
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Identity for $f(x) = \sum_{k=1}^{n}(x+k)^2$

I have a function $$f(x)=(x+1)^2+(x+2)^2 + \dots + (x+n)^2 = \sum_{k=1}^{n}(x+k)^2$$ for some positive integer $n$. I started wondering if there is an equivalent expression for $f(x)$ that can be calculated more directly (efficiently). I began by…
Carser
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