Questions tagged [summation]

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

15
votes
4answers
17k views

Calculating sum of consecutive powers of a number

Here is my problem, I want to compute the $$\sum_{i=0}^n P^i : P\in ℤ_{>1}$$ I know I can implement it using an easy recursive function, but since I want to use the formula in a spreadsheet, is ...
15
votes
3answers
355 views

How to prove that $ \sum\limits_{n=1}^{\infty}n\prod\limits_{k=1}^{n}\frac{1}{1+ka} = \frac{1}{a} $?

Mathematica tells me that $$ \sum_{n=1}^{\infty}n\prod_{k=1}^{n}\frac{1}{1+ka} = \frac{1}{a} $$ I could prove it for $a\rightarrow 0$, $a=1$ and $a\rightarrow \infty$, but could not find a general ...
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 ...
15
votes
4answers
354 views

Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$

Is there an intuitive way, though I am not sure how to find a conceptual proof either, to establish the following identity: $$\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$$ for all natural ...
15
votes
3answers
583 views

Show $\sum_{k=1}^{\infty}\left(\frac{1+\sin(k)}{2}\right)^k$ diverges

Show$$\sum_{k=1}^{\infty}\left(\frac{1+\sin(k)}{2}\right)^k$$diverges. Just going down the list, the following tests don't work (or I failed at using them correctly) because: $\lim \limits_{k\to\...
15
votes
2answers
772 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
2answers
567 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{\...
15
votes
2answers
209 views

If $\,a_n\searrow 0\,$ and $\,\sum_{n=1}^\infty a_n<\infty,\,$ does this imply that $\,n\log n\, a_n\to 0$?

A quite elegant and classic exercise of Calculus (in infinite series) is the following: If the non-negative sequence $\{a_n\}$ is decreasing and $\sum_{n=1}^\infty a_n<\infty$, then $na_n\to 0$. ...
15
votes
2answers
1k views

Is $\lim\limits_{k \to \infty}\left[ \lim\limits_{p \to \infty} \frac{M}{1+3+5+\cdots+ [2^{p(k-1)}-2^{p(k-2)}-2^{p(k-3)}-\cdots-1]}\right]=1$? [closed]

Firstly, my $\LaTeX$, Mathematics and English knowledge is very limited. It is extremely difficult for me to ask this question. Now I am improving myself.I hope you understand me... Look at this ...
15
votes
2answers
503 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 ...
15
votes
2answers
279 views

Evaluate $\sum_{n=1}^\infty 1/n^2$ using $\int_0^1 \int_0^1 \frac{\mathrm{d}x \, \mathrm{d}y}{1-xy}$

This paper http://math.ucsb.edu/~cmart07/Evaluating%20Integrals.pdf hints at a way to compute the sum $$ \sum_{n=1}^\infty \frac{1}{n^2} $$ by expanding it into the double integral $$\int_0^1 \int_0^1 ...
15
votes
3answers
386 views

definite and indefinite sums and integrals

It just occurred to me that I tend to think of integrals primarily as indefinite integrals and sums primarily as definite sums. That is, when I see a definite integral, my first approach at solving it ...
15
votes
0answers
275 views

Geometric representation of Euler-Maclaurin Summation Formula

When reading Tom Apostol's expository article (or the free link), I was expecting more diagrams to come that follow the figure below (or this from the Wolfram project). It was a disappointment not ...
15
votes
0answers
629 views

Proving this binomial identity $\sum_{k=0}^n {n+k \choose k} \frac{1}{2^{k}}= 2^{n}$ [duplicate]

A teacher gave this as a homework question, and I have tried but haven't been able to arrive at a solution. $\sum_{k=0}^n {n+k \choose k} \frac{1}{2^{k}}= 2^{n}$ Could someone prove it, or at least ...
14
votes
11answers
29k views

Prove that $1 + 4 + 7 + · · · + 3n − 2 = \frac {n(3n − 1)}{2}$

Prove that $$1 + 4 + 7 + · · · + 3n − 2 = \frac{n(3n − 1)} 2$$ for all positive integers $n$. Proof: $$1+4+7+\ldots +3(k+1)-2= \frac{(k + 1)[3(k+1)+1]}2$$ $$\frac{(k + 1)[3(k+1)+1]}2 + 3(k+1)-2$$...
14
votes
5answers
538 views

How do we show the equality of these two summations?

How do you show the following? $$\sum \limits_{i=1}^{n}\ \sum \limits_{j=i}^{n}\ \sum \limits_{k=i}^{j}\ 1 = \sum \limits_{j=1}^{n}\ \sum \limits_{k=1}^{j}\ \sum \limits_{i=1}^{k}\ 1 $$ It's not ...
14
votes
2answers
1k views

Summation of an infinite Exponential series

Q. Find the value of - $$ \lim_{n\to\infty} \sum_{r=0}^{n} \frac{2^r}{5^{2^r} +1} $$ My attempt - I seem to be clueless to this problem. Though I think that later terms would be much small and ...
14
votes
8answers
1k views

Truncated alternating binomial sum

It is easily checked that $\displaystyle\sum_{i\ =\ 0}^{n}\left(\, -1\,\right)^{i} \binom{n}{i} = 0$, for example by appealing to the binomial theorem. I'm trying to figure out what happens with the ...
14
votes
3answers
1k views

Can we express $\pi$ in terms of $\sum_{n=1}^\infty\frac1{n^2}$?

Since $$\sum_{n=1}^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6}$$ can we now express $\pi$ in terms of this series by multiplying by $6$ and taking the square root? If not why is this not true? I was ...
14
votes
6answers
750 views

Proving $\sum_{k=1}^{n}{(-1)^{k+1} {{n}\choose{k}}\frac{1}{k}=H_n}$

I've been trying to prove $$\sum_{k=1}^{n}{(-1)^{k+1} {{n}\choose{k}}\frac{1}{k}=H_n}$$ I've tried perturbation and inversion but still nothing. I've even tried expanding the sum to try and find ...
14
votes
3answers
2k views

What do $\{ceps_q\}_{q=0}^Q$ and $\{a_q\}_{q=1}^p$ mean?

As a programmer who hasn't had any higher mathematical training, I sometimes find mathematical equations described in books or online that I'd like to implement in my programs, but they have symbols ...
14
votes
6answers
332 views

show that $\frac{1}{F_{1}}+\frac{2}{F_{2}}+\cdots+\frac{n}{F_{n}}<13$

Let $F_{n}$ is Fibonacci number,ie.($F_{n}=F_{n-1}+F_{n-2},F_{1}=F_{2}=1$) show that $$\dfrac{1}{F_{1}}+\dfrac{2}{F_{2}}+\cdots+\dfrac{n}{F_{n}}<13$$ if we use Closed-form expression $$F_{n}=\...
14
votes
2answers
419 views

A closed form of the double sum $\sum_{m=1}^{\infty}\sum_{n=0}^{m-1}\frac{(-1)^{m-n}}{(m^2-n^2)^2} $

I want to evaluate the double sum $\displaystyle \sum_{m=1}^{\infty}\sum_{n=0}^{m-1}\frac{(-1)^{m-n}}{(m^2-n^2)^2} $ where I know that the value is $-\frac{17\pi^4}{1440}$ , I have failed to evaluate ...
14
votes
2answers
337 views

What is the closed form for $S=\displaystyle\sum_{n=1}^{\infty} \dfrac{\sin ({n})}{n!}$?

How do we find the following sum (closed form)? $$S=\displaystyle\sum_{n=1}^{\infty} \dfrac{\sin ({n})}{n!}$$
14
votes
6answers
753 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 ...
14
votes
2answers
4k views

Finite Sum of Power?

Can someone tell me how to get a closed form for $$\sum_{k=1}^n k^p$$ For $p = 1$, it's just the classic $\frac{n(n+1)}2$. What is it for $p > 1$?
14
votes
4answers
1k views

Combinatorial proof that binomial coefficients are given by alternating sums of squares?

A student recently asked whether there was a combinatorial proof of the following identity: $\begin{equation*} \sum^n_{k=1}(-1)^{n-k}k^2 = {n+1 \choose 2}. \end{equation*}$ I was in a rush and ...
14
votes
3answers
216 views

How to evaluate the sum $\sum_{n=1}^{\infty}\frac{1}{(2n+1)(2n+2)}\left(1+\frac{1}{2}+…+\frac{1}{n}\right)$

How to evaluate the sum: $$S=\sum_{n=1}^{\infty}\frac{1}{(2n+1)(2n+2)}\left(1+\frac{1}{2}+...+\frac{1}{n}\right)$$ Can anyone help me,I really appreciate it.
14
votes
7answers
343 views

How to prove that $\sum_{i=0}^n 2^i\binom{2n-i}{n} = 4^n$.

So I've been struggling with this sum for some time and I just can't figure it out. I tried proving by induction that if the sum above is a $S_n$ then $S_{n+1} = 4S_n$, but I didn't really succeed so ...
14
votes
2answers
366 views

Closed-form of $\sum_{n=0}^\infty\;(-1)^n \frac{\left(2-\sqrt{3}\right)^{2n+1}}{(2n+1)^2\quad}$

The following question is purely my curiosity. During my calculation to answer @Chris'ssis's question in chat room I encountered this series $$\sum_{n=0}^\infty\; \frac{\left(2-\sqrt{3}\right)^{2n+1}}{...
14
votes
3answers
489 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..?
14
votes
2answers
12k views

Value of Summation of $\log(n)$

Context: I am learning Dijstra's Algorithm to find shortest path to any node, given the start node. Here, we can use Fibonnacci Heap as Priority Queue. Following is few lines of algorithm: ...
14
votes
4answers
1k views

Recent Question in American Math Monthly, proposed by Donald Knuth

Problem 11985, by Donald Knuth, American Mathematical Monthly, June-July, 2017: For fixed $s,t \in \mathbb{N}$. with $s\leq t$. let $a_{n}=\sum\limits_{k=s}^{t}$ $ {n}\choose{k}$. Prove that this ...
14
votes
1answer
13k views

Can we express sum of products as product of sums?

I've got an expression which is sum of products like: $$a_1 a_2 + b_1 b_2 + c_1 c_2 + \cdots,$$ but the real problem I'm solving could be easily solved if I could convert this expression into ...
14
votes
2answers
395 views

A conjecture including binomial coefficients

Question: $$\sum_{k=1}^{n}k\binom{2n}{n+k}=\frac n2\binom{2n}{n}$$ is true for every $n\in \mathbb N$? If this is true, then how can we prove this? When I was playing with numbers, I conjectured ...
14
votes
2answers
535 views

Prove that ${\sum\limits_{n=1}^{\infty}}(-1)^{n-1} \frac{H_n}{n} = \frac{\pi^2}{12} - \frac12\ln^2 2$

We know that $H_n = \sum_{j=1}^{n}{1 \over j}$. Article in The Sum of Certain Series Related To Harmonic Numbers of Omran Kolba, we have proof of this identity which involves some advanced concepts. ...
14
votes
1answer
674 views

Infinite Series $\sum\limits_{n=1}^{\infty}\frac{(-1)^n}{n}\left\lfloor\frac{\log(n)}{\log(2)}\right\rfloor$

How to prove that $$\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\left\lfloor\frac{\log(n)}{\log(2)}\right\rfloor=\gamma$$ Can we find a known value for $\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\left\lfloor\frac{\...
14
votes
2answers
437 views

Another Series $\sum\limits_{k=2}^\infty \frac{\log(k)}{k}\sin(2k \mu \pi)$

I ran across an interesting series in a paper written by J.W.L. Glaisher. Glaisher mentions that it is a known formula but does not indicate how it can be derived. I think it is difficult. $$\sum_{k=...
14
votes
2answers
463 views

Efficient computation of $\sum_{k=1}^n \lfloor \frac{n}{k}\rfloor$

I realize that there is probably not a closed form, but is there an efficient way to calculate the following expression? $$\sum_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor$$ I've noticed $$\sum_{k=...
14
votes
2answers
485 views

Proving that $\sum_{n=1}^\infty \frac{\sin\left(n\frac{\pi}{3}\right)}{(2n+1)^2}=\frac{G}{\sqrt 3} -\frac{\pi^2}{24}$

Trying to show using a different approach that $\int_0^1 \frac{\sqrt x \ln x}{x^2-x+1}dx =\frac{\pi^2\sqrt 3}{9}-\frac{8}{3}G\, $ I have stumbled upon this series: $$\sum_{n=1}^\infty \frac{\sin\...
14
votes
3answers
294 views

Sum of $\{n\sqrt{2}\}$

I would like to prove (rigorously, not intuitively) that $$\sum_{n=1}^N \{n\sqrt{2}\}=\frac{N}{2}+\mathcal{O}(\sqrt{N})$$ where $\{\}$ is the "fractional part" function. I understand intuitively why ...
14
votes
2answers
217 views

Prove that $\sum_{n=1}^\infty \left(\phi-\frac{F_{n+1}}{F_{n}}\right)=\frac{1}{\pi}$

So, I know that $$\lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\phi$$ where $F_n$ stands for the n'th Fibonacci number I was interested in measuring the error of the convergence of the above limit and was ...
14
votes
1answer
381 views

Show all roots of $\sum_{k=0}^n 2^{k(n-k)} x^k$ are real (December 6, 2014 Putnam problem)

Show that for each positive integer n, all roots of the polynomial $\sum_{k=0}^n 2^{k(n-k)} x^k$ are real numbers. I have no idea where to start. From this year's Putnam, problem B4.
14
votes
3answers
390 views

Sum of the first integer powers of $n$ up to k

Pascal's triangle has a lot of interesting patterns in it; one of which is the triangular numbers and their extensions. Mathematically: $$\sum_{n=1}^k1=\frac{k}{1}$$ $$\sum_{n=1}^kn=\frac{k}{1}\cdot\...
14
votes
2answers
853 views

Help with proving a statement based on Riemann sums?

Suppose we have the original Riemann sum with no removed partitions, where $f(x)$ is continuous and reimmen integratable on the closed interval $[a,b]$. $$\lim_{n\to\infty}\sum_{i=1}^{n}f\left(a+\left(...
14
votes
3answers
353 views

Why isn't finite calculus more popular?

I'm reading through Concrete Math, and learning about finite calculus. I've never heard of it anywhere else, and a Google search found very few relevant sources. It seems to me an incredibly powerful ...
13
votes
9answers
3k views

Has anyone heard of this maths formula and where can I find the proof to check my proof is correct? $\sum^n_{i = 1}i + \sum^{n-1}_{i=1}i = n^2$

The formula basically is: The sum of all integers before and including $n$, plus all the integers up to and including $n-1$. This will find $n^2$. $$ \sum^n_{i = 1}i + \sum^{n-1}_{i=1}i = n^2 $$
13
votes
7answers
2k views

How would I express the series $|1+1+1|+|1+1-1|+|1-1+1|+|1-1-1|+|-1+1+1|+|-1+1-1|+|-1-1+1|+|-1-1-1|$ in summation notation?

I tried putting the series $$ |1+1+1|+|1+1-1|\\+|1-1+1|+|1-1-1|\\+|-1+1+1|+|-1+1-1|\\+|-1-1+1|+|-1-1-1| $$ into Wolfram Alpha and typing "in summation notation" but it wouldn't tell me what it is in ...
13
votes
6answers
6k views

Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction

I am just starting out learning mathematical induction and I got this homework question to prove with induction but I am not managing. $$\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$$ ...
13
votes
5answers
2k views

Sum of an infinite series $(1 - \frac 12) + (\frac 12 - \frac 13) + \cdots$ - not geometric series?

I'm a bit confused as to this problem: Consider the infinite series: $$\left(1 - \frac 12\right) + \left(\frac 12 - \frac 13\right) + \left(\frac 13 - \frac 14\right) \cdots$$ a) Find the sum $S_n$ ...