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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Finding a general expression for $\sum_{k=1}^n a_{k} \exp\left(\frac{ick}{n}\right)$?

Consider the sum $$ \sum_{k=1}^n a_{k} \exp\left(\frac{ick}{n}\right). $$ I have heard of methods that treat exponential sums. I was wondering if it's possible to find general expressions for exponential sums with "weights". Note that $a_{k}$ in…
Avyakta Purush
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How does one determine that $\sum_{k=1}^{n} \frac{1}{(n+k)(n+k+1)}$ evaluates to $\frac{n}{2n^2+3n+1}$?

How does one determine that $$\sum_{k=1}^{n} \frac{1}{(n+k)(n+k+1)}$$ evaluates to $$\frac{n}{2n^2+3n+1}\ ?$$ What are the simplification steps involved?
Thicc
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Finding a specific series sum $\sum_{k=1}^{i-1}(\frac{((ak^3+a(1-2i+2n)k^2+2b)}{2ak})^{-1}$

I am stuck in this specific sum equation as below. Sum it $k$ is from $1$ to $i-1$ and others are static variables. Please help to find $\sum_{k=1}^{i-1}(\frac{((ak^3+a(1-2i+2n)k^2+2b)}{2ak})^{-1}$ I appreciated your helps. Thank you.
JongH
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Can this equation be solved for $N_{mpn}$?

I am trying to solve for $N_{mpn}$. Can this be solved? $$\sum_{i=1}^K\frac{V_id_iP_i}{1-e^{-V_id_iN_{mpn}}}=\sum_{i=1}^KV_id_in_i$$ In researching this, it did not appear that I could pull any constants out because every variable is part of the…
kbennett
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What is the result of this sum, for big values of n?

The sum is the following: $$\sum_{i=0}^{n-1} \left[i\frac{2^i}{2^n-1}\right]$$ I know that the result of this sum, for big values of $n$, should be $n-2$, but I am not aware of the procedure used to get to this solution.
XSword24
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Explain how $\sum_{k = 1}^n\frac{k^2}{2^{k - 1}} = \sum_{k = 0}^{n - 1}\frac{\left(k + 1\right)^2}{2^k}$?

Can someone explain simply why $\displaystyle\sum_{k = 1}^n\dfrac{k^2}{2^{k - 1}} = \displaystyle\sum_{k = 0}^{n - 1}\dfrac{\left(k + 1\right)^2}{2^k}$? I don't get why we go from $k$ to $k+1$ in the numerator, please help me understand it.
radarGlimpse
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Can't simplify this summation

Here's the question: $\sum_{i=1}^{n} \frac{1}{i^2 + i}$ How do you express this summation in terms of n? I'm at a loss of where to go with this. I know the formulas for the summation of $i^2$ and $i$, but I'm not sure where to use these (or if…
Stoodent
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Switching summation indices

This seems to be really basic and might be a case of begging the question here, but I can't seem to prove the that the right hand side implies the left hand side. Let $J$ be the set of all $j$ and $K(j)$ be the set of all $k$ that are valid for each…
Jia Cheng Sun
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To justify whether or not jointly Gaussian

Let $X_{1}$,..., $X_{n}$ be random variables, and define $$Y_{k} := \sum_{i=1}^{k} X_{i}, k = 1,...,n.$$ Suppose that $Y_{1}$,..., $Y_{n}$ are jointly Gaussian. Determine whether or not $X_{1}$,...,$X_{n}$ are jointly Gaussian. I do not know how to…
givan
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Adding two summations confusion

This week, a teacher of mine wrote this down. He's been wrong before and so I've been looking at it and trying to see how this is true, but I can't seem to see how left leads to right. Is this correct? $$\sum_{i=0}^{n} b_{2i}x^{2i} + \sum_{i=1}^{n}…
Wishiknew
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Simplification of Summation

I am trying to simplify or analysis the convergence of the following equation. $c_k$ is between $0$ and $1$. Can someone please give an idea for that? $$ \frac{\Bigg(\sum_{k=1}^{K} \frac{2c_k}{a_k^2+b_k^2}\Bigg)^2}{\Bigg(\sum_{k=1}^{K}…
user70822
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Sum $\sum_{k=0}^{\infty} \frac{k^2}{2^k}$

How to go about summing $\sum_{k=0}^{\infty} \frac{k^2}{2^k}$.
sbp
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summing equality over all (i,j) pairs

$$(φ_j − φ_i )J_{ji} = (ψ_j − ψ_i )I_{ji} ⋯(1)$$ $$\sum_j φ_j\sum_i J_{ji} - \sum_i φ_i\sum_j J_{ji} = \sum_j ψ_j\sum_i I_{ji} - \sum_i ψ_i\sum_j I_{ji}⋯(2)$$ This is related to reciprocity theorem in circuits. Just taking $\sum_j $ leaves…
scitamehtam
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Why does $\sum_{n=0}^{N-1} e^{ikna} e^{-ilna} = N\delta_{kl}$?

Why does $\sum_{n=0}^{N-1} e^{ikna} e^{-ilna} = N\delta_{kl}$? I tried to solve this question but I could not really proceed. This summation sign is making this difficult for me to understand.
Tulika Sharma
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Why is $ \frac{m}{n} \sum\limits_{i=0}^{n-1} \frac{1}{m-i} = \frac{1}{n/m} \sum\limits_{k=m-n+1}^{m} \frac{1}{k} $

I don't understand why $$ \frac{m}{n} \sum_{i=0}^{n-1} \frac{1}{m-i} $$ is equal to $$ \frac{1}{n/m} \sum_{k=m-n+1}^{m} \frac{1}{k} $$ Can someone explain this please?
NimaJan
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