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

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65
votes
17answers
4k views

Proving the identity $\sum\limits_{k=1}^n {k^3} = {\Large(}\sum\limits_{k=1}^n k{\Large)}^2$ without induction

I recently proved that $$ \sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2 $$ Using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial ...
32
votes
20answers
7k views

Proof that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$?

I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals: $$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$ I really ...
32
votes
1answer
649 views

Finding the sum- $x+x^{2}+x^{4}+x^{8}+x^{16}\cdots$

If $S = x+x^{2}+x^{4}+x^{8}+x^{16}\cdots$ Find S. Note:This is not a GP series.The powers are in GP. My Attempts so far: 1)If $S(x)=x+x^{2}+x^{4}+x^{8}+x^{16}\cdots$ Then ...
31
votes
4answers
599 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 ...
30
votes
4answers
561 views

$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$

Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction. $$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
29
votes
2answers
632 views

If $\left(1^a+2^a+\cdots+n^{a}\right)^b=1^c+2^c+\cdots+n^c$ for some $n$, then $(a,b,c)=(1,2,3)$?

Question : Is the following conjecture true? Conjecture : Let $a,b(\ge 2),c,n(\ge 2)$ be natural numbers. If $$\left(\sum_{k=1}^nk^a\right)^b=\sum_{k=1}^nk^c\ \ \ \ \ \cdots(\star)$$ for some $n$, ...
28
votes
3answers
3k views

The sum of an uncountable number of positive numbers

Claim:If $(x_\alpha)_{\alpha\in A}$ is a collection of real numbers $x_\alpha\in [0,\infty]$ such that $\sum_{\alpha\in A}x_\alpha<\infty$, then $x_\alpha=0$ for all but at most countably many ...
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 ...
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: ...
25
votes
2answers
4k views

$\sum k! = 1! +2! +3! + \cdots + n!$ ,is there a generic formula for this?

I came across a question where I needed to find the sum of the factorials of the first $n$ numbers. So I was wondering if there is any generic formula for this? Like there is a generic formula for ...
24
votes
6answers
974 views

Evaluating $‎\sum_{n=2}^{\infty}\frac{\zeta(n)}{k^n}$

‎If $f\left(z \right)=\sum_{n=2}^{\infty}a_{n}z^n$ and $\sum_{n=2}^{\infty}|a_n|$ converges then‎, $$\sum_{n=1}^{\infty}f\left(\frac{1}{n}\right)=\sum_{n=2}^{\infty}a_n\zeta\left(n\right)‎.$$ ‎Since ...
24
votes
1answer
563 views

Proving $\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$

Wikipedia informs me that $$S = \vartheta(0;i)=\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$$ I tried considering $f(x,n) = e^{-x n^2}$ so that its ...
23
votes
9answers
2k views

why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$

why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$? I know this is well-known. But how to prove it rigorously? Even mathematical induction does not seem so straight-forward. ...
22
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} ...
22
votes
5answers
655 views

How find this sum $\sum\limits_{n=0}^{\infty}\frac{1}{(3n+1)(3n+2)(3n+3)}$

Find this sum $$I=\sum_{n=0}^{\infty}\dfrac{1}{(3n+1)(3n+2)(3n+3)}$$ My try: let $$f(x)=\sum_{n=0}^{\infty}\dfrac{x^{3n+3}}{(3n+1)(3n+2)(3n+3)},|x|\le 1$$ then we have ...
21
votes
21answers
12k views

Proof for formula for sum of sequence $1+2+3+\ldots+n$?

Apparently $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$. How? What's the proof? Or maybe it is self apparent just looking at the above? Does this problem have a name and maybe a presence on the net? ...
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 ...
20
votes
3answers
579 views

Finding the value of $\sum\limits_{k=0}^{\infty}\frac{2^{k}}{2^{2^{k}}+1}$

Does this weighted sum of reciprocals of Fermat numbers, $$ F=\sum_{k=0}^{\infty}\dfrac{2^{k}}{2^{2^{k}}+1} $$ have a nice closed form? Wolfram says it's $1$. Thanks.
20
votes
2answers
318 views

How prove that:$\varphi(2)+\varphi(3)+\varphi(4)+\cdots+\varphi(n)\ge\frac{n(n-1)}{4}+1$

Prove that for $n\ge 3$, $$\varphi(2)+\varphi(3)+\varphi(4)+\cdots+\varphi(n)\ge\dfrac{n(n-1)}{4}+1$$ where $\varphi$ is the Euler's totient function I think we must use this ...
19
votes
7answers
1k views

Is there a formula for $\sum_{n=1}^{k} \frac1{n^3}$?

I am searching for the value of $$\sum_{n=k+1}^{\infty} \frac1{n^3} \stackrel{?}{=} \sum_{n = 1}^{\infty} \frac1{n^3} - \sum_{n=1}^{k} \frac1{n^3} = \zeta(3) - \sum_{n=1}^{k} \frac1{n^3}$$ For which ...
19
votes
2answers
365 views

How to either prove or disprove if it is possible to arrange a series of numbers such the sum of any two adjacent number adds up to a prime number

I'm wondering if it's possible to write a theorem to prove or disprove the possibility of arranging a sequence of numbers (1,2,...n) such that the sum of any two numbers adds up to a prime number. An ...
19
votes
3answers
443 views

Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n}$

Please help me to find a closed form for the sum $$\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n},$$ where $H_n$ are harmonic numbers: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$
19
votes
4answers
493 views

How close can $\sum_{k=1}^n \sqrt{k}$ be to an integer?

How close can $S(n) = \sum_{k=1}^n \sqrt{k}$ be to an integer? Is there some $f(n)$ such that, if $I(x)$ is the closest integer to $x$, then $|S(n)-I(S(n))|\ge f(n)$ (such as $1/n^2$, $e^{-n}$, ...). ...
19
votes
1answer
797 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: ...
18
votes
6answers
2k views

Induction: $\sum_{k=0}^n \binom nk k^2 = n(1+n)2^{n-2}$

I found crazy (for me at least) induction example, in fact it just would be nice to prove. (Even have problems with starting) Any hints are highly valued: ...
18
votes
6answers
514 views

Is there a fundamental reason that $\int_b^a = -\int_a^b$

Is there a fundamental reason that switching the order of the limits in an integral results in the negative, i.e., $$\int_b^af(x)\,dx = -\int_a^bf(x)\,dx?$$ As far as I can tell, this is just chosen ...
18
votes
2answers
444 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 ...
18
votes
4answers
306 views

Find the smallest k such that $n^k > \sum_{i=0}^{n-1} i^k$

Let $n \in \mathbb{N}$. Is it possible to find the smallest $k \in \mathbb{N}$ such that $$n^k > \sum_{i=1}^{n-1} i^k \ ?$$ It's easy to prove that such $k$ exist because: $$n^k > 1^k + 2^k ...
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 ...
17
votes
4answers
2k views

Is there a way to denote the calculation $1+2+3+\dots+n$? [duplicate]

Since $n!$ represents $$1\cdot2\cdot3\cdots n,$$ I am wondering if there is a way to represent $$1+2+3+\dots+n?$$ What are some usual notations for the computation of some common sequences? Any other ...
17
votes
3answers
446 views

Counting binary sequences with no more than $2$ equal consecutive numbers

I invented the following problem, but I can't solve it. There are $n$ $1$'s and $n$ $0$'s and my question is what is the number of ways to arrange them avoiding $3$ equal consecutive numbers. So for ...
17
votes
1answer
730 views

Prove that sum is finite

Let $j \in \mathbb{N}$. Set $$ a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!} $$ and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$. Please help me to prove that the following sum is ...
16
votes
3answers
389 views

How prove this sum $\sum_{n=1}^{\infty}\binom{2n}{n}\frac{(-1)^{n-1}H_{n+1}}{4^n(n+1)}$

show that $$\sum_{n=1}^{\infty}\binom{2n}{n}\dfrac{(-1)^{n-1}H_{n+1}}{4^n(n+1)}=5+4\sqrt{2}\left(\log{\dfrac{2\sqrt{2}}{1+\sqrt{2}}}-1\right)$$ where ...
16
votes
1answer
508 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.
16
votes
4answers
545 views

Proving that $\sum_{i=0}^{n}\binom{n}{i}i^{n-i}(n-i)^{i}\le\frac{1}{2}n^n$

How can we prove that $$\displaystyle\sum_{i=0}^{n}\binom{n}{i}i^{n-i}(n-i)^{i}\le\dfrac{1}{2}n^n$$ where $\displaystyle\binom{n}{i}=\dfrac{n!}{i!(n-i)!}$. This inequality is very interesting. I ...
16
votes
1answer
216 views

Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^a H_n}{2^n}$

Is there a closed form for the sum $$\sum_{n=1}^\infty\frac{(-1)^n n^a H_n}{2^n},$$ where $H_n$ are harmonic numbers: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$ This is a ...
16
votes
2answers
223 views

Conjecture: the sequence of sums of all consecutive primes contains an infinite number of primes

Starting from 2, the sequence of sums of all consecutive primes is: $$\begin{array}{lcl}2 &=& 2\\ 2+3 &=& 5 \\ 2+3+5 &=& 10 \\ 2+3+5+7 &=& 17 \\ ...
15
votes
4answers
475 views

How find this sum $\sum_{ab+cd=2^m}ac=?$

Find this sum (where $m$ is a fixed positive integer) $$\sum_{\substack{ab+cd=2^m\\ a,b,c,d \text{ are odd}}}ac.$$ My idea: since $$ab+cd=2^m\Longrightarrow ab=2^m-cd$$ and $a,b,c,d$ is odd ...
15
votes
4answers
571 views

Find the infinite sum $\sum_{n=1}^{\infty}\frac{1}{2^n-1}$

How to evaluate this infinite sum? $$\sum_{n=1}^{\infty}\frac{1}{2^n-1}$$
15
votes
2answers
580 views

What is the max of $n$ such that $\sum_{i=1}^n\frac{1}{a_i}=1$ where $2\le a_1\lt a_2\lt\cdots\lt a_n\le 99$?

What is the max of $n$ such that $$\sum_{i=1}^n\frac{1}{a_i}=1$$ where $a_{i}\ (i=1,2,\cdots,n)$ are integers which satisfy $2\le a_1\lt a_2\lt\cdots\lt a_n\le 99$ ? Also, I need how to prove that ...
15
votes
1answer
289 views

How find this $\sum\limits_{n=0}^{\infty}(-1)^{n}\left(\frac{1}{n+1}-\frac{1}{n+2}+\frac{1}{n+3}-\cdots\right)^{2}$

Find the value $$\sum\limits_{n=0}^{\infty}(-1)^{n}\left(\frac{1}{n+1}-\frac{1}{n+2}+\frac{1}{n+3}-\cdots\right)^{2}$$ This problem is from ...
15
votes
2answers
554 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 ...
14
votes
11answers
2k views

Can the factorial function be written as a sum?

I know of the sum of the natural logarithms of the factors of n! , but would like to know if any others exist.
14
votes
8answers
2k views

How to determine equation for $\sum_{k=1}^n k^3$

How do you find an algebraic formula for $\sum_{k=1}^n k^3$? I am able to find one for $\sum_{k=1}^n k^2$, but not $k^3$. Any hints would be appreciated.
14
votes
2answers
522 views

Intuitive ways to get formula of cubic sum

Is there an intuitive way to get cubic sum? From this post: combination of quadratic and cubic series and Wikipedia: Faulhaber formula, I get $$1^3 + 2^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$$ I think ...
14
votes
5answers
916 views

Proving the identity $\sum_{n=-\infty}^\infty e^{-\pi n^2x}=x^{-1/2}\sum_{n=-\infty}^\infty e^{-\pi n^2/x}.$

Can you help prove the functional equation: $$\sum_{n=-\infty}^\infty e^{-\pi n^2x}=x^{-1/2}\sum_{n=-\infty}^\infty e^{-\pi n^2/x}.$$ Specifically, I am looking for a solution using complex ...
14
votes
2answers
461 views

Sum with binomial coefficients and a square root

I encountered this sum from working on an integral: $$\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\sqrt{k}$$ I don't think it can be written as a hypergeometric function, because of this square root. Does ...
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 ...
14
votes
1answer
352 views

Show that $ \sum_{r=1}^{n-1}\binom{n-2}{r-1}r^{r-1}(n-r)^{n-r-2}= n^{n-2} $

Show that $$ \sum_{r=1}^{n-1}\binom{n-2}{r-1}r^{r-1}(n-r)^{n-r-2}= n^{n-2} $$ I don't know whether such identity already exists, or has been posted here before. I discovered this identity while ...