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 understand this double sum of probabilities

Can someone please explain how this double sum works and why the LHS = RHS? $$2 \sum_{1\leq j < k \leq N} P(F_j F_k) = \sum_{j\neq k} P(F_j F_k)$$
user477465
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Sum of increasing powers

In the answer to a different question someone wrote: Let $\omega = e^{2 \pi i / n}$ which implies $\omega^n = 1$. $$ 1 + \omega + \omega^2 + \ldots + \omega^{n-1} = \frac{\omega^n-1}{\omega-1} = 0 $$ I tried to understand this for at least two…
Boris Month
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Infinite sum of $\frac{x^n}{n+1}$

I was solving a calculus problem and found myself stuck trying to find to solve: $$S = \sum_{n=0}^{\infty} \frac{x^n}{n+1}$$ Wolfram Alpha says: But I have no idea how this sum is evaluated. Any help is appreciated.
João Vítor G. Lima
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Summation Rules Mistake?

I'm looking at a solution for a stats problem and I think there is a mistake. The solution is as follows: $$m_i^{-1} \sum_{e=1}^{m_i} u_{i,e} = m_i^{-1}\sum_{e=1}^{m_i}(f_i + u_{i,e}) = f_i + m_{i}^{-1}\sum_{e=1}^{m_i}v_{i,e}$$ I believe there was a…
Nicklovn
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Evaluate $\sum ^n _{r=0} \binom{n}{r} \tan^{2r}\left(\frac \pi 3 \right)$

Evaluate $$\sum ^n _{r=0} \binom{n}{r} \tan^{2r}\left(\frac \pi 3 \right)$$ So I've got to a point at which I don't know how to go any further, any help would be appreciated. My workings so far are shown. $$\sum ^n _{r=0} \frac…
H.Linkhorn
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Solving a summation using summation rules

I am given two problems. $$\sum_{i=0}^{n} (4i + \frac{3}{4}n + \frac{1}{2}) \tag 1$$ $$ \sum_{i=1}^{n} (4i + \frac{3}{4}n + \frac{1}{2}) \tag 2$$ I am asked to solve it. I know I can manipulate this into three separate summations: Attempt: $$…
Arthur Green
  • 167
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How to determine the general formula for a summation?

What are some strategies for trying to determine the general formula for a summation? For example, let's say I'm trying to determine the general formula for $\sum_{i=1}^n \frac{i}{n}$ I was not sure how to approach this question, so I plugged and…
James Smithson
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A (probably easy) question about exchanging the order of this double sum

I would like to exchange the order of the following: $\sum_{k=1}^{i-1}\sum_{r=1}^{2k}$ (stuff). I feel like it should be easy, but so far I am only able to produce $\sum_{r=1}^{2(i-1)}\sum_{k=??}^{??}$ (stuff). In particular, I don't see how to…
MathIsArt
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Summation with power

Please help me with the below Summation $\sum_{r=0}^{\infty}2^r \times n^{\frac{1}{2^r}}$ While solving problem based on finding time complexity, I came upto this expression. I am unable to move further. Please help.
user3767495
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Is there a closed form of $\sum_{k=1}^{\infty}\frac{\sin (2k-1)\pi x}{(2k-1)^5\pi^5}$

Is there a closed form of $$\sum_{k=1}^{\infty}\frac{\sin (2k-1)\pi x}{(2k-1)^5\pi^5}$$ The above sum if a solution of some problem so I need to simplify it to get a clear equation if possible .......thanks
omarbaker100
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Sum of distance traveled between two runners in summation form

Problem: Assume two runners $A$ and $B$ are $2h_0$ meters apart. They start running towards each other at a speed of $a$ meters per second. Assume there is a third runner that runs in between the two runners at a speed of $b$ meters per second, with…
B.Li
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Multiplication of sums

In this set of online math notes the following is stipulated: $$\left(\sum_{n=0}^\infty a_n\right)\left(\sum_{n=0}^\infty b_n\right) =\sum_{n=0}^\infty (c_n) $$ Where apparently $c_n = \sum_{i=0}^n a_ib_{n-i}$. Replacing $c_n$ into above equation…
user32882
  • 702
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$\sum_{n = 1}^{D - 1} \frac{n}{D - n}$ written as a function of $D$?

$$\sum_{n = 1}^{D - 1} \frac{n}{D - n}$$ Is it possible to reduce this summation to a function of $D$?
bestscammer5
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How do we define summation over an arbitrary index set including negative values?

I just read that for $f: I \rightarrow [0, \infty),$ we can define $\sum_{i \in I} f(i)$ as $\sup\{\sum_{i \in F} f(i) \mid F \subset I, F$ finite $\}.$ It makes sense that we can define it in this way. It coincides with my notion for countable…
green frog
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Why is the Big O characterizaion of this summation O(n^(n+1))?

If we have this summation $$ \sum_{k=1}^n n^k $$ The time complexity of this function is O($n^{n+1}$). Why would that be? If I'm not mistaken this is a function which is the addition of n to all powers between 1 and n, meaning the term with the…
M Zein
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