# Questions tagged [summation]

Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.

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### Find the expected value of $\frac{1}{X+1}$ where $X$ is binomial

The problem: X is a binomial random variable, find $E[\frac{1}{X+1}]$ n and p are not given PDF for a binomial distribution is $\binom{n}{k}p^k(1-p)^{n-k}$ Expected value is $\sum{x_ip(x_i)}$ But ...
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### Proving $\sum_{k=1}^n k\cdot k! = (n+1)!-1$ without using mathematical Induction. [duplicate]

Possible Duplicate: Summation of a factorial This equation is given: $$1\cdot1! + 2\cdot2! + 3\cdot3! + \ldots + n\cdot n! = (n+1)! - 1$$ I've solved it using mathematical induction but I'm ...
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### Proof by induction that $\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$

Prove via induction that $\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$ Having a very difficult time with this proof, have done pages of work but I keep ending up with 1/(k+2). Not sure when to ...
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### Binomial Identity $\sum\binom{2n+1}{2k+1}\binom{m+k}{2n} = \binom{2m}{2n}$

I'm looking for a reference with the proof of the following binomial identity: $$\sum_{k=0}^n \binom{2n+1}{2k+1}\binom{m+k}{2n} = \binom{2m}{2n}$$ I've looked in a number of textbooks that have a ...
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### Combinatorial interpretation of an alternating binomial sum

Let $n$ be a fixed natural number. I have reason to believe that $$\sum_{i=k}^n (-1)^{i-k} \binom{i}{k} \binom{n+1}{i+1}=1$$ for all $0\leq k \leq n.$ However I can not prove this. Any method to prove ...
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### Find the value of $\left[\frac{1}{\sqrt 2}+\frac{1}{ \sqrt 3}+…+\frac{1}{\sqrt {1000}}\right]$

Find the value of $$\left[\frac{1}{\sqrt 2}+\frac{1}{\sqrt 3}+.....+\frac{1}{\sqrt {1000}}\right]$$.Where [•] denote the greatest integer function. I am very confused about this problem. I tried to ...
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### Show that $\sum_{n=2}^m \binom{n}{2} = \binom{m+1}{3}$

I need a hand in showing that $$\sum_{n=2}^m \binom{n}{2} = \binom{m+1}{3}$$ Thanks in advance for any help.
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### Double sum trouble

Evaluate: $$\sum_{j=1}^{\infty} \sum_{i=1}^{\infty} \frac{j^2i}{3^j(j3^i+i3^j)}$$ Honestly, I don't see where to start with this. I am sure that this is a trick question and I am missing something ...
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### How do you find the value of $\sum_{r=0}^{44} \tan^2(2r+1)$?

Problem: Find the value of $$\sum_{r=0}^{44} \tan^2(2r+1)$$ Note: The angles here are in degrees. I don't know how to solve this question because trigonometric simplifications didn't get me ...
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I'm trying to find the sum of : $$\sum_{i=1}^n\frac i {2^i}$$ I've tried to run $i$ from $1$ to $∞$ , and found that the sum is $2$ , i.e : $$\sum_{i=1}^\infty\frac i {2^i} =2$$ since : $$(1/2 +... 5answers 941 views ### How Are the Solutions for Finite Sums of Natural Numbers Derived? So, I've been learning set theory on my own (Lin, Shwu-Yeng T., and You-Feng Lin. Set Theory: An Intuitive Approach. Houghton Mifflin Co., 1974.) and have come across infinite sums of natural numbers. ... 5answers 8k views ### Show \sum\limits_{d|n}\phi(d) = n. [duplicate] Show \sum\limits_{d|n}\phi(d) = n. Example : \sum\limits_{d|4}\phi(d) = \phi(1) + \phi(2) + \phi(4) = 1 + 1 + 2 = 4 I was told this has a simple proof. Problem is, I can not think of a way to ... 6answers 699 views ### Race of the wealthy gamblers: How do I get this closed form? UPDATE: If you can prove the following identity:$$\sum\limits_{t=0}^p \left(\frac{{2t+1 \choose t}}{2^{2t+1}}\right)^2 \frac{1}{t+2} = -1 + \frac{(2+p)}{2^{4p+4}}{2p+3 \choose p+1}^2$$Then this ... 3answers 930 views ### Show this equality (The factorial as an alternate sum with binomial coefficients). Why does the following equality hold?$$n!=\sum_{k=1}^n (-1)^{n-k} \binom{n}{k} k^n$$3answers 10k views ### How to prove Lagrange trigonometric identity [duplicate] I would to prove that$$1+\cos \theta+\cos 2\theta+\ldots+\cos n\theta =\displaystyle\frac{1}{2}+ \frac{\sin\left[(2n+1)\frac{\theta}{2}\right]}{2\sin\left(\frac{\theta}{2}\right)}$$given that$$1+...
I want to find this $$\sum_{k=1}^n \gcd(k,n)$$ but I don't know how to solve. Does anybody can help me to finding this problem. Thanks.