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

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19
votes
1answer
805 views

Solving $x^k+(x+1)^k+(x+2)^k+\cdots+(x+k-1)^k=(x+k)^k$ for $k\in\mathbb N$

Letting $k$ be a natural number, can we solve the following $k$-th degree equation ? $$x^k+(x+1)^k+(x+2)^k+\cdots+(x+k-1)^k=(x+k)^k\ \ \ \ \cdots(\star).$$ The following two are famous: ...
11
votes
5answers
415 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 ...
16
votes
1answer
509 views

Simplify $\left({\sum_{k=1}^{2499}\sqrt{10+{\sqrt{50+\sqrt{k}}}}}\right)\left({\sum_{k=1}^{2499}\sqrt{10-{\sqrt{50+\sqrt{k}}}}}\right)^{-1}$

Simplify $$\frac{\sum_{k=1}^{2499}\sqrt{10+{\sqrt{50+\sqrt{k}}}}}{\sum_{k=1}^{2499}\sqrt{10-{\sqrt{50+\sqrt{k}}}}}$$ I don't have any good idea. I need your help.
14
votes
1answer
1k views

Combinatorial proof of a Fibonacci identity: $n F_1 + (n-1)F_2 + \cdots + F_n = F_{n+4} - n - 3.$

Does anyone know a combinatorial proof of the following identity, where $F_n$ is the $n$th Fibonacci number? $$n F_1 + (n-1)F_2 + \cdots + F_n = F_{n+4} - n - 3$$ It's not in the place I thought it ...
9
votes
7answers
365 views

Why does $\sum_{k=1}^{\infty}\dfrac{{\sin(k)}}{k}={\dfrac{\pi-1}{2}}$?

Inspired by this question (and far more straightforward, I am guessing), Mathematica tells us that $$\sum_{k=1}^{\infty}\dfrac{{\sin(k)}}{k}$$ converges to $\dfrac{\pi-1}{2}$. Presumably, this can ...
6
votes
2answers
120 views

How find this $\sum_{n=1}^{\infty}\frac{(-1)^{n-1}\zeta_{n}(3)}{n}=?$

Question: show that $$\sum_{n=1}^{\infty}\dfrac{(-1)^{n-1}\zeta_{n}(3)}{n}=\dfrac{19\pi^4}{1440}-\dfrac{3}{4}\zeta{(3)}\ln{2}?$$ where $$\zeta_{n}(3)=\sum_{k=1}^{n}\dfrac{1}{k^3}$$ But I use ...
5
votes
3answers
742 views

Proving that $\sum_{k=0}^{n} {{m+k} \choose{m}} = { m+n+1 \choose m+1 }$

I have to prove that: $$\sum_{k=0}^{n} {{m+k} \choose{m}} = { m+n+1 \choose m+1 }$$ I tried to open up the right side with Pascal's definition that: $$ { n \choose k} = {n-1 \choose {k}} + {n-1 ...
4
votes
3answers
1k views

Solve $\sum nx^n$

I am trying to find a closed form solution for $\sum_{n\ge0} nx^n\text{, where }\lvert x \rvert<1$. This solution makes sense to me: $\sum_{n\ge0} x^n=(1-x)^{-1} \\ \frac{d}{d x} \sum_{n\ge0} x^n ...
3
votes
5answers
4k views

calculate sum of square of first n odd numbers

Is there an analytical expression for the summation $$1^2+3^2+5^2+\cdots+(2n-1)^2$$ and how did you derive it?
0
votes
3answers
206 views

$1+2+3+4+5+… = -\frac{1}{12}$. Is there any intuition for this? [duplicate]

I was looking into a Numberphile video here. The guy says he was unable to find an intuition. Does there exist one? Is the premise, $1-1+1-1+...=\frac{1}{2}$, reasonable mathematically?
0
votes
2answers
153 views

How to simplify this summation containing floor

Here's the summation: $$ S = \sum_{i=1}^{n}\left\lfloor\frac{n}{i}\right\rfloor $$ It can also be written as $$ S = \sum_{i=1}^{n}\left\lfloor\frac{n- (n \mod i)}{i}\right\rfloor $$ The answer will ...
4
votes
1answer
369 views

sum of $\displaystyle \frac{\sin nx }{n^4}$

Consider : $\displaystyle f(x)= \sum_{n=1}^{\infty} \frac{\sin nx }{n^4}$ Find : $\displaystyle \int_0^{x} f(t)\ \mathrm{d}t$.
2
votes
2answers
213 views

Double Summation: Need help to handle $ i \neq j $ : $ \sum_{i=0 \to 7,\ j=1 \to 8,\ i\neq j} (8i + j) $

[Q1]. Can I ? ( write the same summation as ) : $$ \sum_{i=0, i \neq j}^7 \sum_{j=1}^8 (8i + j) \tag{1}$$ I tried to solve the following Summation as follows: Let i = m-1 then, $ \sum_{i=0,\ i ...
2
votes
3answers
107 views

How would you prove $\sum_{i=1}^{n} (3/4^i) < 1$ by induction?

How would you prove this by induction? $\sum_{i=1}^{n} (3/4^i) < 1$, $ \forall n \geq 2$ I can do the base case but don't know how to to finish it
0
votes
1answer
156 views

Is every sum to something a limit of the sum?

Old title: Is it always true that a sum to some $m$ equals the limit of that sum to some $x$ such that $x$ tends to $m$? $\ $ This may seem a stupid question, but I'm placing it because although ...
-1
votes
1answer
92 views

Use Mathematical Induction to prove that $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} +…+\frac{1}{n(n+1)}=1-\frac{1}{n+1}$

Use Mathematical induction to prove that for all integers, $n$ is greater than or equal to $1$. I am confused on what to do after I do the the basis step that is using $n$ as $1$. $$\frac{1}{1 \cdot ...
31
votes
4answers
612 views

$\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor$ is true?

I found the following relational expression by using computer: For any natural number $n$, $$\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor.$$ Note ...
28
votes
1answer
529 views

Geometric interpretation for sum of fourth powers

Summing the first $n$ first powers of natural numbers: $$\sum_{k=1}^nk=\frac12n(n+1)$$ and there is a geometric proof involving two copies of a 2D representation of $(1+2+\cdots+n)$ that form a ...
18
votes
2answers
445 views

How does one [easily] calculate $\sum\limits_{n=1}^\infty\frac{\mathrm{pop}(n)}{n(n+1)}$?

How does one [easily] calculate $\sum\limits_{n=1}^\infty\frac{\mathrm{pop}(n)}{n(n+1)}$, where $\mathrm{pop}(n)$ counts the number of bits '1' in the binary representation of $n$? Is there any trick ...
15
votes
2answers
555 views

How to evaluate $\int_{0}^{1}{\frac{{{\ln }^{2}}\left( 1-x \right){{\ln }^{2}}\left( 1+x \right)}{1+x}dx}$

I want to evaluate $$\int_{0}^{1}{\frac{{{\ln }^{2}}\left( 1-x \right){{\ln }^{2}}\left( 1+x \right)}{1+x}dx}$$ I run this integral on Maple, It does converge. How we get a closed form? Is that ...
18
votes
4answers
437 views

How find this sum $I_n=\sum_{k=0}^{n}\frac{H_{k+1}H_{n-k+1}}{k+2}$

$$I_n=\sum_{k=0}^{n}\dfrac{H_{k+1}H_{n-k+1}}{k+2}$$ where $$H_{n}=1+\dfrac{1}{2}+\cdots+\dfrac{1}{n}$$ my try:since ...
11
votes
2answers
299 views

Is $\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}$ true for $m\in\mathbb N$?

Question : Is the following true for any $m\in\mathbb N$? $$\begin{align}\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}\qquad(\star)\end{align}$$ Motivation : I reached ...
21
votes
5answers
1k views

Sum of the form $r+r^2+r^4+\dots+r^{2^k} = \sum_{i=1}^k r^{2^k}$

I am wondering if there exists any formula for the following power series : $$S = r + r^2 + r^4 + r^8 + r^{16} + r^{32} + ...... + r^{2^k}$$ Is there any way to calculate the sum of above series (if ...
8
votes
4answers
404 views

Sum : $\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^3}$

Prove that : $$\sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2k+1)^3}=\frac{\pi^3}{32}.$$ I think this is known (see here), I appreciate any hint or link for the solution (or the full solution).
23
votes
4answers
2k views

Identity involving Euler's totient function: $\sum \limits_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}$

Let $\varphi(n)$ be Euler's totient function, the number of positive integers less than or equal to $n$ and relatively prime to $n$. Challenge: Prove $$\sum_{k=1}^n \left\lfloor \frac{n}{k} ...
26
votes
1answer
522 views

Evaluating the sum $\lim_{n\to \infty}\sqrt[2]{2+\sqrt[3]{2+\sqrt[4]{2+\cdots+\sqrt[n]{2}}}}$

The following nested radical $$\lim_{n\to \infty}\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}$$ is known to converge to 2. We can consider a similar nested radical where the degree of the radicals increases: ...
5
votes
3answers
307 views

Closed form for the sum of even fibonacci numbers?

I recently took a look a project euler, and I am trying to think of a smart way to do number 2. I looked at the sequence, and I saw that the question is basically asking for $$ \sum_{i=1}^n F_{3i} $$ ...
4
votes
2answers
179 views

Integral of the Von karman equation

What is the result of this integral, and how can I proceed: $$ \int_{-\infty}^{\infty}{c_{1} \over\left(1 + c_{2}\,x^{2}\right)^{5/6}}\, \cos\left(x\tau\right)\,{\rm d}x\,,\qquad c_{1}, ...
11
votes
2answers
544 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 ...
5
votes
1answer
237 views

Sum involving the hypergeometric function, power and factorial functions

I am finding some trouble in calculating the following sum involving the hypergeometric function, power and factorial functions. $$ \sum_{y=1}^\infty \mathrm{e}^z \cdot {}_1F_1\left(1-y;2;-z\right) ...
5
votes
1answer
1k views

Show $1 + 2 \sum_{n=1}^N \cos n x = \frac{ \sin (N + 1/2) x }{\sin \frac{x}{2}}$ for $x \neq 0$ [duplicate]

For $x \neq 0$, $$ 1 + 2 \sum_{n=1}^N \cos n x = \frac{ \sin (N + 1/2) x }{\sin \frac{x}{2}} $$
4
votes
2answers
123 views

Euler-Maclaurin summation for $e^{-x^2}$

I want to approximate the sum $$\sum_{k=0}^\infty e^{-k^2}$$ using the Euler-Maclaurin formula $$\sum_{k=0}^\infty f(k) = \int_0^\infty f(x) \, dx + \frac{1}{2}(f(0) + f(\infty)) + ...
3
votes
2answers
164 views

Summation involving totient function: $\sum_{d\mid n} \varphi(d)=n$ [duplicate]

Prove that:$$\sum_{d\mid n} \varphi(d)=n$$ Where $\varphi(n)$ denotes the number of positive integers $m$ less than or equal to $n$ such that $\gcd(m,n)=1$ I am lost here, any help would be ...
2
votes
1answer
73 views

A double sum with combinatorial factors

Let $n$, $p$ and $j$ be integers. As a byproduct of some other calculations I have discovered the following identity: \begin{equation} \sum\limits_{p=0}^{j} \sum\limits_{p_1=0}^j \binom{p+p_1}{p_1} ...
2
votes
1answer
88 views

evaluating sum of limits [duplicate]

I was given this problem in definite integral and limits chapter $$\lim_{n \to \infty}\left( \frac{n}{n^2 + 1^2}+ \frac{n}{n^2 + 2^2} + \frac{n}{n^2 + 3^2 }+........+\frac{n}{n^2 + n^2}\right)$$ which ...
1
vote
3answers
369 views

Show that $f_0 - f_1 + f_2 - \cdots - f_{2n-1} + f_{2n} = f_{2n-1} - 1$ when $n$ is a positive integer

Letting $f_n$ be the Fibonacci numbers, show that $f_0 - f_1 + f_2 - \cdots - f_{2n-1} + f_{2n} = f_{2n-1} - 1$ when $n$ is a positive integer. Just some homework help. Need to prove. Thank you in ...
6
votes
2answers
99 views

Sum Involving Bernoulli Numbers : $\sum_{r=1}^n \binom{2n}{2r-1}\frac{B_{2r}}{r}=\frac{2n-1}{2n+1}$

How can we prove that $$\sum_{r=1}^n \binom{2n}{2r-1}\frac{B_{2r}}{r}=\frac{2n-1}{2n+1}$$ where $B_{2r}$ are the Bernoulli numbers? $$\begin{array}{c|c|c|} n & \frac{2n-1}{2n+1} & ...
5
votes
6answers
2k views

Proving $\sum_{k=0}^{n}k{n\choose k}^2 = n{2n-1 \choose n-1} $

[Corrected question] I'm struggling at proving the following combinatorical identity: $$\sum_{k=0}^{n}k{n\choose k}^2 = n{2n-1 \choose n-1} $$ I would like to see a combinatorical (logical) solution, ...
5
votes
2answers
110 views

Sum up to number $N$ using $1,2$ and $3$

So the question asked was finding out the number of ways(combinations), a given number $N$ can be formed using the sum of $1,2$ or $3$. (eg) For $n = 8$, the answer is $10$ The given solution for ...
4
votes
2answers
88 views

Prove: $\sum_{k<n, (k,n)=1} k= \frac{1}{2}n \varphi (n)$

Prove: $\sum_{k<n, (k,n)=1}k = \frac{1}{2}n \varphi (n)$ I have had strep throat and missed the lecture discussing properties of the Euler function. Any help in solving this is appreciated. ...
3
votes
3answers
187 views

Given n $\in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$

I tried to solve it using induction, but that got me no were, in the middle of the equation stat appearing ks that I don't see how to get out of the equation. I think the easiest way to prove it, it's ...
3
votes
5answers
182 views

$\sum_{k=1}^nH_k = (n+1)H_n-n$. Why?

This is motivated by my answer to this question. The Wikipedia entry on harmonic numbers gives the following identity: $$ \sum_{k=1}^nH_k=(n+1)H_n-n $$ Why is this? Note that I don't just ...
2
votes
2answers
192 views

Alternating sum of binomial coefficients $\sum(-1)^k{n\choose k}\frac{1}{k+1}$ [duplicate]

I would appreciate if somebody could help me with the following problem Q:Calculate the sum: $$ \sum_{k=1}^n (-1)^k {n\choose k}\frac{1}{k+1} $$
1
vote
2answers
59 views

Prove $r^n - s^n = (r-s)\sum_{j=0}^{n-1} r^js^{n-j-1}$ by induction

Prove $$r^n - s^n = (r-s)\sum_{j=0}^{n-1} r^js^{n-j-1} $$ $(1)$by induction. I've verified that $$n=1: r^1 - s^1 = (r-s)(r^0s{1-0-1}) = r-s$$ Assume $(1)$ is true for $n \le k$. That is $$r^k - s^k ...
0
votes
3answers
94 views

Derive Closed form sum of N^2

Can anyone explain to me how you would derive this ? I have this question asked in a CS class and can't figure out how to derive it. it has to be derived as you would with sum of N ex ...
8
votes
1answer
477 views

Apéry's constant ($\zeta(3)$) value

I tried to find some proofs about the Apéry's constant, but I didn't find any intuitive proof. Is this constant given by the "brutal force" summing of $1 + \frac{1}{2^3} + \frac{1}{3^3} + ...
4
votes
3answers
89 views

Factorial identity $n!=1+(1-1/1!)n+(1-1/1!+1/2!)n(n-1)+\cdots$

Show that $\displaystyle{n!=1+\left(1-\frac1{1!}\right)n+\left(1-\frac1{1!}+\frac1{2!}\right)n(n-1)+\cdots}$. I can't figure out how this can be solved. I tried to use the binomial theorem but I ...
4
votes
2answers
68 views

Finding the summation of the floor of the series identity

I would appreciate if somebody could help me with the following problem: Q: How to proof ? The number of positive divisors of $n$ is denoted by $d(n)$ ...
4
votes
2answers
3k views

Compute $1^2 + 3^2+ 5^2 + \cdots + (2n-1)^2$ by mathematical induction

I am doing mathematical induction. I am stuck with the question below. The left hand side is not getting equal to the right hand side. Please guide me how to do it further. $1^2 + 3^2+ 5^2 + ...
3
votes
1answer
200 views

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

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 ...