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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8
votes
2answers
28k views

Sum of $\cos(k x)$ [duplicate]

I'm trying to calculate the trigonometric sum : $$\sum\limits_{k=1}^{n}\cos(k x)$$ This is what I've tried so far : $$\renewcommand\Re{\operatorname{Re}} \begin{align*} \sum\limits_{k=1}^{n}\cos(k x) ...
5
votes
2answers
2k views

Finite Summation of Fractional Factorial Series

Is there a closed form solution for the following series? (Without Using Gamma Function): $$ S=\sum _{i=1}^{n-1} \frac{1}{(i+1)!} $$
9
votes
2answers
372 views

How to find $\lfloor 1/\sqrt{1}+1/\sqrt{2}+\dots+1/\sqrt{100}\rfloor $ without a calculator?

$$ \left\lfloor\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} +\dots+ \frac{1}{\sqrt{100}}\right\rfloor =\,? $$ I rationalized the denominator and then I think I should somehow group the numbers, but i don'...
6
votes
5answers
2k views

Difficulties in a proof by mathematical induction (involves evaluating $\sum r3^r$).

Please help. I've been stuck on this for 2 days. Haven't found any easy explaining text. The question is: Prove by mathematical induction that : $$ \sum_{r=1}^n r3^r = \frac{3}{4} \left[ 3^n \...
1
vote
4answers
14k views

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 ...
10
votes
5answers
788 views

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 ...
9
votes
4answers
184 views

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 ...
5
votes
2answers
551 views

Euler-Maclaurin Summation

Using EM summation formula estimate $$ \sum_{k=1}^n \sqrt k $$ up to the term involving $\frac{1}{\sqrt n}$ My attempt is $$ \sum_{k=1}^n \sqrt k = \frac{2 \sqrt{n^3}}{3} -\frac{2}{3} + \frac 1 ...
21
votes
0answers
933 views

Are there unique solutions for $n=\sum_{j=1}^{g(k)} a_j^k$?

Edward Waring, asks whether for every natural number $n$ there exists an associated positive integer s such that every natural number is the sum of at most $s$ $k$th powers of natural numbers (...
2
votes
5answers
328 views

Help with proof using induction: $1 + \frac{1}{4} + \frac{1}{9}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$ [duplicate]

I am having trouble with the following proof: For every positive integer $n$: $$1 + \frac{1}{4} + \frac{1}{9}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$$ My work: I have tried to add $\frac{1}{(k+1)^...
2
votes
2answers
82 views

Show convergence and calculate the limit: $\lim \limits_{n \to \infty}\sum \limits_{k=n}^{2n}\frac{k}{k^2+n^2}$ [duplicate]

I guess it has something to do with Riemann sums but this is new for me. $\displaystyle\lim \limits_{n \to \infty}\sum \limits_{k=n}^{2n}\frac{k}{k^2+n^2}$ How do I start?
0
votes
5answers
3k views

Prove by induction: $2^n = C(n,0) + C(n,1) + \cdots + C(n,n)$ [duplicate]

This is a question I came across in an old midterm and I'm not sure how to do it. Any help is appreciated. $$2^n = C(n,0) + C(n,1) + \cdots + C(n,n).$$ Prove this statement is true for all $n \...
47
votes
1answer
1k views

Why does this ratio of sums of square roots equal $1+\sqrt2+\sqrt{4+2\sqrt2}=\cot\frac\pi{16}$ for any natural number $n$?

Why is the following function $f(n)$ constant for any natural number $n$? $$f(n)=\frac{\sum_{k=1}^{n^2+2n}\sqrt{\sqrt{2n+2}+{\sqrt{n+1+\sqrt k}}}}{\sum_{k=1}^{n^2+2n}\sqrt{\sqrt{2n+2}-{\sqrt{n+1+\...
15
votes
2answers
571 views

Prove $\sum_{n=1}^{\infty}\arctan{\left(\frac{1}{n^2+1}\right)}=\arctan{\left(\tan\left(\pi\sqrt{\frac{\sqrt{2}-1}{2}}\right)\cdots\right)}$

show that: $$\sum_{n=1}^{\infty}\arctan{\left(\dfrac{1}{n^2+1}\right)}=\arctan{\left(\tan\left(\pi\sqrt{\dfrac{\sqrt{2}-1}{2}}\right)\cdot\dfrac{e^{\pi\sqrt{\dfrac{\sqrt{2}+1}{2}}}+e^{-\pi\sqrt{\...
23
votes
2answers
2k views

Integrating the formula for the sum of the first $n$ natural numbers

I was messing around with some math formulas today and came up with a result that I found pretty neat, and I would appreciate it if anyone could explain it to me. The formula for an infinite ...
21
votes
5answers
907 views

Proving $\sum_{k=0}^{2m}(-1)^k{\binom{2m}{k}}^3=(-1)^m\binom{2m}{m}\binom{3m}{m}$ (Dixon's identity)

I found the following formula in a book without any proof: $$\sum_{k=0}^{2m}(-1)^k{\binom{2m}{k}}^3=(-1)^m\binom{2m}{m}\binom{3m}{m}.$$ I don't know how to prove this at all. Could you show me how ...
19
votes
5answers
36k views

Sum of Square roots formula.

I would like to know if there is formula to calculate sum of series of square roots $\sqrt{1} + \sqrt{2}+\dotsb+ \sqrt{n}$ like the one for the series $1 + 2 +\ldots+ n = \frac{n(n+1)}{2}$. Thanks in ...
17
votes
3answers
621 views

Show that $\sum\limits_{k=0}^n\binom{2n}{2k}^{\!2}-\sum\limits_{k=0}^{n-1}\binom{2n}{2k+1}^{\!2}=(-1)^n\binom{2n}{n}$

How can I prove the identity: $$ \sum_{k=0}^n\binom{2n}{2k}^2-\sum_{k=0}^{n-1}\binom{2n}{2k+1}^2=(-1)^n\binom{2n}{n}? $$ Maybe, can we expand $$ f(x)=(1+x)^{2n}? $$ Thank you.
14
votes
3answers
491 views

How to find sums like $\sum_{k=0}^{39} \binom{200}{5k}$

How do I find sums like these?-- $$S=\displaystyle\sum_{k=0}^{39} \dbinom{200}{5k}$$ that is, when there is a summation of binomial coefficients, but with jumps of some terms..?
13
votes
5answers
24k views

sum of this series: $\sum_{n=1}^{\infty}\frac{1}{4n^2-1}$

I am trying to calculate the sum of this infinite series after having read the series chapter of my textbook: $$\sum_{n=1}^{\infty}\frac{1}{4n^2-1}$$ my steps: $$\sum_{n=1}^{\infty}\frac{1}{4n^2-1}=...
15
votes
3answers
1k views

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 ...
14
votes
6answers
755 views

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 ...
8
votes
4answers
786 views

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 ...
13
votes
12answers
1k views

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.
13
votes
2answers
785 views

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 ...
10
votes
2answers
296 views

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 ...
10
votes
3answers
14k views

Finite Sum $\sum_{i=1}^n\frac i {2^i}$

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 +...
9
votes
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. ...
8
votes
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 ...
16
votes
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 ...
8
votes
3answers
930 views

Show this equality (The factorial as an alternate sum with binomial coefficients).

Why does the following equality hold? $$n!=\sum_{k=1}^n (-1)^{n-k} \binom{n}{k} k^n$$
13
votes
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+...
11
votes
1answer
6k views

Sum of GCD(k,n)

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.
4
votes
3answers
1k views

Fibonacci Numbers Proof: $ f_n = \binom n0 + \binom{n-1}1 +\dots+ \binom{n-k}k$

Prove the following fibonacci sequence, which appear in Pascal's Triangle. I am not sure where to start on this, any pointers? $$ f_n = {n\choose0} + {n-1\choose1} + ... + {n-k\choose k}$$ where $\...
3
votes
4answers
305 views

Gosper's Identity $\sum_{k=0}^n{n+k\choose k}[x^{n+1}(1-x)^k+(1-x)^{n+1}x^k]=1 $

The page on Binomial Sums in Wolfram Mathworld http://mathworld.wolfram.com/BinomialSums.html (Equation 69) gives this neat-looking identity due to Gosper (1972): $$\sum_{k=0}^n{n+k\choose k}[x^{n+1}(...
3
votes
1answer
2k views

sum of the product of consecutive legendre symbols is -1

How do I prove the formula $\newcommand{\jaco}[2]{\left(\frac{#1}{#2}\right)}\sum\limits_{a=1}^{p-2} \jaco{a(a+1)}p = -1$ where a varies from 1 to p-2 and p is a prime I got as far as $\jaco{p-a}p = \...
3
votes
2answers
5k views

Asymptotic estimate of the sequence of harmonic series $\sum_{k=1}^{n} \frac{1}{k} $ [closed]

Asymptotic estimate of the sequence of partial sums, for $n\rightarrow \infty$: $$ s_n=\sum_{k=1}^{n} \frac{1}{k} $$
10
votes
2answers
1k views

Alternative combinatorial proof for $\sum\limits_{r=0}^n\binom{n}{r}\binom{m+r}{n}=\sum\limits_{r=0}^n\binom{n}{r}\binom{m}{r}2^r$

I have a question with regards to combinatorics. I am supposed to show the following combinatorial identity: $\sum\limits_{r=0}^n\binom{n}{r}\binom{m+r}{n}=\sum\limits_{r=0}^n\binom{n}{r}\binom{m}{r}2^...
6
votes
4answers
473 views

Combinatorial Proof for Binomial Identity: $\sum_{k = 0}^n \binom{k}{p} = \binom{n+1}{p+1}$ [duplicate]

I am studying combinatorics and I came across the identity $$\sum\limits_{k=0}^n \binom kp =\binom {n+1}{p+1}.$$ I have read the algebraic proof and it does not appeal to me. Is there an elegant ...
6
votes
6answers
186 views

Find the value of $\sum_{k=1}^{n}k\binom{n}{k}$?

Find the value of $\sum_{k=1}^{n}k\binom{n}{k}$ ? I know that $\sum_{k=0}^{n}\binom{n}{k}= 2^{n}$ and so, $\sum_{k=1}^{n}\binom{n}{k}= 2^{n}-1$ but how to deal with $k$ ?
6
votes
4answers
691 views

Evaluate $\sum_{k=1}^nk\cdot k!$

I discovered that the summation $\displaystyle\sum_{k=1}^n k\cdot k!$ equals $(n+1)!-1$. But I want a proof. Could anyone give me one please? Don't worry if it uses very advanced math, I can just ...
5
votes
2answers
782 views

Fibonacci combinatorial identity: $F_{2n} = {n \choose 0} F_0 + {n\choose 1} F_1 + … {n\choose n} F_n$ [duplicate]

Can someone explain how to prove the following identity involving Fibonacci sequence $F_n$ $F_{2n} = {n \choose 0} F_0 + {n\choose 1} F_1 + ... {n\choose n} F_n$ ?
4
votes
5answers
489 views

Proving the total number of subsets of S is equal to $2^n$

Student here! Just reading Liebecks Introduction to pure mathematics for fun and I made an attempt at proving the total number of subsets of S is equal to $2^n$. I realized that the total number of ...
3
votes
6answers
7k views

Sum of nth roots of unity

Question: If $c\neq 1$ is an $n^{th}$ root of unity then, $1+c+...+c^{n-1} = 0$ Attempt: So I have established that I need to show that $$\sum^{n-1}_{k=0} e^{\frac{i2k\pi}{n}}=\frac{1-e^{\frac{ik2\pi}...
7
votes
8answers
6k views

How do I come up with a function to count a pyramid of apples?

My algebra book has a quick practical example at the beginning of the chapter on polynomials and their functions. Unfortunately it just says "this is why polynomial functions are important" and moves ...
6
votes
4answers
2k views

Sum of the first $n$ triangular numbers - induction

Question: Prove by mathematical induction that $$(1)+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)=\frac{1}{6}n(n+1)(n+2)$$ is true for all positive integers n. Attempt: I did the the induction steps and I ...
6
votes
4answers
1k views

How to prove that $\sum_{k=0}^n \binom nk k^2=2^{n-2}(n^2+n)$ [duplicate]

I know that $$\sum_{k=0}^n \binom nk k^2=2^{n-2}(n^2+n),$$ but I cannot find a way how to prove it. I tried induction but it did not work. On wiki they say that I should use differentiation but I do ...
4
votes
6answers
6k views

Prove by induction $3+3 \cdot 5+ \cdots +3 \cdot 5^n = \frac{3(5^{n+1} -1)}{4}$

My question is: Prove by induction that $$3+3 \cdot 5+ 3 \cdot 5^2+ \cdots +3 \cdot 5^n = \frac{3(5^{n+1} -1)}{4}$$ whenever $n$ is a nonnegative integer. I'm stuck at the basis step. If I ...
4
votes
6answers
2k views

Evaluating even binomial coefficients

Can someone give me a hint how to evaluate $$ \binom{n}{0}+\binom{n}{2}+\cdots+\binom{n}{o(n)},$$ where $o(n)$ is $n$ if $n$ is even and $n-1$ otherwise ?
3
votes
1answer
808 views

Evaluate $\sum_{k = 0}^{n} {n\choose k} k^m$ [duplicate]

So, I wonder what is the evaluation of $$\sum_{k = 0}^{n} {n\choose k} k^m\text{,}\qquad (*)$$ where $m,n\in \mathbb{N}$. One of my tries: knowing that $$k^m = \sum_{j = 0}^{m}\text{S}(m,j)\cdot k(k-...