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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Sum of odd numbers, greater than zero, generates a new term. How?

$$\sum_{\text{odd }k} \frac{-2e^{-ikt}}{ik\pi} = \sum_{\text{odd }k>0} \frac{-2e^{-ikt}+2e^{ikt}}{ik\pi}$$ How does limiting all $k$ values to greater than zero, introduce the new term in the numerator? I've been staring at this for an hour and I…
AnimationB
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Simplify $\sum_{i=0}^{n-1} \sum_{j=i}^{n-1}1$

Question: Simplify $$\sum_{i=0}^{n-1} \sum_{j=i}^{n-1}1$$ My attempt: $$\sum_{i=0}^{n-1} \sum_{j=i}^{n-1}1 =\sum_{i=0}^{n-1} (n-1-i) $$. Would this now be the sum of the first $n-1$ integers?
user444945
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Is there a simple formula for computing this sum?

I was watching this video from James Grime on his channel singingbanana: https://www.youtube.com/watch?v=LejoPGtliTs and I wondered how many times does he have to flip every othelo piece. So I found this sum: $$\sum_{k=1}^n \left \lfloor \frac{n}{k}…
pavle
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Interpretation of double sum

I found the following equation in a paper: $$\alpha=\sum\limits_{i=l}^2\sum\limits_{l=1}^2(-1)^l\frac{\alpha_i\left[hf-E_g(T)+(-1)^lk\theta_i\right]}{\left(\exp\left[(-1)^l\frac{\theta_i}{T}\right]-1\right)}$$ Now I am wondering how to interpret…
arc_lupus
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Generating a certain number after $k$ operations. Obtain a formula for this number as a function of $k$

There are n cookies on a table. Adam did this series of steps: In the 1st step he put 1 cookie in the middle of every two neighbouring cookies, in the 2nd step he put 2 cookies in the middle of every two neighbouring cookies, in the kth step he put…
pavle
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What is $\sum_{j=i}^{n}j$

How do I calculate $$\sum_{j=i}^{n}j?$$ WolframAlpha gave me $-\frac{1}{2}(i-n-1)(i+n)$ but I don't understand how it got there
PTN
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Is there an equation for the sum of alternating cubes?

The following sequences for sum of alternating cubes: Odd cubes: [1, 28, 153, 496, 1225, 2556, 4753, 8128, 13041, 19900, 29161, 41328, 56953, 76636, 101025, 130816, 166753, 209628, 260281, 319600] Even cubes: [8, 72, 288, 800, 1800, 3528, 6272,…
KaliMa
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Splitting a summation into even and odd components

I'm having trouble understanding how the the second line of the splitting of this summation works. I don't understand how the sums of each individual parts can still run from n = 0 to infinity without over-counting and making the sum…
falsovuoto
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Problem with simplifying summation

I have $$-2[\sum_{i=1}^n (y_i*(\bar y)) - \sum_{i=1}^n (y_i*(\bar x)*\frac {S_{(xy)}}{S_{(xx)}})+\sum_{i=1}^n (y_i*x_i*\frac {S_{(xy)}}{S_{(xx)}})]$$ This is a term of a larger sum I am dealing with to derive something in regression. I am allowed to…
the boy 88
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find the precise range of of the following sum

I need to find the precise range of the following sum to infinity: $$\sum_{n=1}^\infty\frac{x^n}{4^nn}$$ Can someone post how to do it because I have no idea, I'm also given that it converges...
mc1
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What is $\sum_{n=1}^k\frac{1}{n^3-c}$?

I am currently doing a proof which requires me, given some constant $c$ and some constant $k$, to evaluate the following: $$\sum_{n=1}^k\frac{1}{n^3-c}$$ Is there any known way to do such? If so, can it be generalized…
user477899
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Evaluate the sum $\sum_{n=1}^{b}n\binom{a}{n}\binom{b}{n}$

Let $a$ and $b$ be natural numbers such that $a \geq b \geq 1$. How can we evaluate the following sum? $$ \sum_{n=1}^{b}n\binom{a}{n}\binom{b}{n}$$
David E.
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Difference of $n$-th power of two consecutive integers

This question came about after reading this other question here. Show that $$\sum_{r=1}^n a^{r-1}\left[\binom n{r-1}-(a+1)^{n-r}\right]=0$$ without expanding the summation in full. If we expand the summation in full we…
Hypergeometricx
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Evaluating sums with n inside using perturbation method

The perturbation method is useful for evaluating sums like $\sum_{k=0}^{n}{a_k}$, but every example I've found is a case where $a_k$ is a sequence dependent only upon the index variable $k$. It is explained that we set $S_n=\sum_{k=0}^{n}{a_k}$ and…
Joald
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Clarification: Summation step

I am reading Concrete Mathematics by Donald Knuth. There is a method to solve summation, particularly for $\sum_{k=0}^n k^2$, called Expand and contract. I don't understand just a single step. i.e. $$\sum_{1\leq k \leq n} k^2 = \sum_{1\leq j \leq k…
Abhisek
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