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Questions tagged [summation]

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

302
votes
8answers
30k views

The length of toilet roll

Fun with Math time. My mom gave me a roll of toilet paper to put it in the bathroom, and looking at it I immediately wondered about this: is it possible, through very simple math, to calculate (with ...
149
votes
23answers
12k views

Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ 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 interpretation ...
127
votes
5answers
41k views

How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?

How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? For example here is the sum of $\cos$ series: $$\sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n \times \frac{...
113
votes
31answers
57k views

Sum of First $n$ Squares Equals $\frac{n(n+1)(2n+1)}{6}$

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 ...
111
votes
5answers
28k views

Value of $\sum\limits_n x^n$

Why does the following hold: \begin{equation*} \displaystyle \sum\limits_{n=0}^{\infty} 0.7^n=\frac{1}{1-0.7} = 10/3 ? \end{equation*} Can we generalize the above to $\displaystyle \sum_{n=0}^{\...
98
votes
24answers
14k views

Can an infinite sum of irrational numbers be rational?

Let $S = \sum_ {k=1}^\infty a_k $ where each $a_k$ is positive and irrational. Is it possible for $S$ to be rational, considering the additional restriction that none of the $a_k$'s is a linear ...
93
votes
5answers
22k 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 $\...
90
votes
14answers
36k views

How to find a general sum formula for the series: 5+55+555+5555+…?

I have a question about finding the sum formula of n-th terms. Here's the series: $5+55+555+5555$+...... What is the general formula to find the sum of n-th terms? My attempts: I think I need to ...
83
votes
30answers
164k 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? PS: This problem is know as "The sum of the first $n$ positive ...
71
votes
7answers
9k views

Multiple-choice: sum of primes below $1000$

I sat an exam 2 months ago and the question paper contains the problem: Given that there are $168$ primes below $1000$. Then the sum of all primes below 1000 is (a) $11555$ (b) $76127$ (c) $...
66
votes
4answers
6k views

Identity for convolution of central binomial coefficients: $\sum\limits_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$

It's not difficult to show that $$(1-z^2)^{-1/2}=\sum_{n=0}^\infty \binom{2n}{n}2^{-2n}z^{2n}$$ On the other hand, we have $(1-z^2)^{-1}=\sum z^{2n}$. Squaring the first power series and comparing ...
65
votes
3answers
7k views

Review of my T-shirt design [closed]

I'm a graphics guy and a wanna-be mathematician. Is the T-shirt design below okay? Or if there's a bone headed error, I'd appreciate a heads up. `
60
votes
6answers
45k views

Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$

I was hoping to find a more "mathematical" proof, instead of proving logically $\displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n \choose n}$. I already know the logical Proof: $${n \choose k}^2 = {...
57
votes
4answers
44k views

What is the term for a factorial type operation, but with summation instead of products? [duplicate]

(Pardon if this seems a bit beginner, this is my first post in math - trying to improve my knowledge while tackling Project Euler problems) I'm aware of Sigma notation, but is there a function/name ...
57
votes
12answers
32k views

Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction

How can I prove that $$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$ for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction. Thanks
56
votes
7answers
7k views

Can we calculate $ i\sqrt { i\sqrt { i\sqrt { \cdots } } }$?

It might be obvious that $2\sqrt { 2\sqrt { 2\sqrt { 2\sqrt { 2\sqrt { 2\sqrt { \cdots } } } } } } $ equals $4.$ So what about $i\sqrt { i\sqrt { i\sqrt { i\sqrt { i\sqrt { i\sqrt { \cdots } } } } } } ...
54
votes
3answers
53k 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 ...
54
votes
1answer
3k views

Is $\sqrt1+\sqrt2+\dots+\sqrt n$ ever an integer?

Related: Can a sum of square roots be an integer? Except for the obvious cases $n=0,1$, are there any values of $n$ such that $\sum_{k=1}^n\sqrt k$ is an integer? How does one even approach such a ...
51
votes
4answers
5k views

What is the explanation for this visual proof of the sum of squares?

Supposedly the following proves the sum of the first-$n$-squares formula given the sum of the first $n$ numbers formula, but I don't understand it.
51
votes
6answers
2k views

Generalizing the sum of consecutive cubes $\sum_{k=1}^n k^3 = \Big(\sum_{k=1}^n k\Big)^2$ to other odd powers

We have, $$\sum_{k=1}^n k^3 = \Big(\sum_{k=1}^n k\Big)^2$$ $$2\sum_{k=1}^n k^5 = -\Big(\sum_{k=1}^n k\Big)^2+3\Big(\sum_{k=1}^n k^2\Big)^2$$ $$2\sum_{k=1}^n k^7 = \Big(\sum_{k=1}^n k\Big)^2-3\Big(\...
50
votes
15answers
7k views

How can you prove that $1+ 5+ 9 + \cdots +(4n-3) = 2n^{2} - n$ without using induction?

Using mathematical induction, I have proved that $$1+ 5+ 9 + \cdots +(4n-3) = 2n^{2} - n$$ for every integer $n > 0$. I would like to know if there is another way of proving this result ...
50
votes
2answers
2k views

Xmas Maths 2015

Simplify the expression below into a seasonal greeting using commonly-used symbols in commonly-used formulas in maths and physics. Colours are purely ornamental! $$ \begin{align} \frac{ \color{green}{...
48
votes
2answers
10k 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 $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+\...
47
votes
4answers
2k 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 ...
46
votes
11answers
4k views

What is the total sum of the cardinalities of all subsets of a set?

I'm having a hard time finding the pattern. Let's say we have a set $$S = \{1, 2, 3\}$$ The subsets are: $$P = \{ \{\}, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}, \{1, 2, 3\} \}$$ And the ...
45
votes
7answers
13k views

Is it possible to write a sum as an integral to solve it?

I was wondering, for example, Can: $$ \sum_{n=1}^{\infty} \frac{1}{(3n-1)(3n+2)}$$ Be written as an Integral? To solve it. I am NOT talking about a method for using tricks with integrals. But ...
44
votes
13answers
8k views

Proof of the Hockey-Stick Identity: $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$

After reading this question, the most popular answer use the identity $$\sum_{t=0}^n \binom{t}{k} = \binom{n+1}{k+1}.$$ What's the name of this identity? Is it the identity of the Pascal's triangle ...
43
votes
4answers
3k 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} \right\...
42
votes
6answers
6k views

If $f(x)=\frac{1}{x^2+x+1}$, how to find $f^{(36)} (0)$?

If $f(x)=\frac{1}{x^2+x+1}$, find $f^{(36)} (0)$. So far I have tried letting $a=x^2+x+1$ and then finding the first several derivatives to see if some terms would disappear because the third ...
41
votes
3answers
935 views

Calculate the following infinite sum in a closed form $\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$?

Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction. $$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
41
votes
4answers
1k 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 ...
38
votes
8answers
4k 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. ...
37
votes
6answers
6k views

Why is this 'Proof' by induction not valid?

I am trying to understand why induction is valid. For instance why would this 'proof' not be valid under the principle of proof by induction ? : $$ \sum_{k=1}^{\infty} \frac{1}{k} \lt \infty$$ ...
37
votes
6answers
2k views

Finite Sum $\sum\limits_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}$

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 $(\star)$ by ...
36
votes
9answers
3k views

Is difference of two consecutive sums of consecutive integers (of the same length) always square?

I am an amateur who has been pondering the following question. If there is a name for this or more information about anyone who has postulated this before, I would be interested about reading up on it....
36
votes
1answer
4k 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 $$S(x)-S(x^{2})=x$$ ...
36
votes
3answers
1k views

Interesting representation of $e^x$

So I discovered the following formula by using the Taylor series for $\ln (x+1)$ $$x= \ln (x+1)+\frac{1}{2}\ln(x^2+1)-\frac{1}{3}\ln(x^3+1)+\frac{1}{2}\ln(x^4+1)-\frac{1}{5}\ln(x^5+1)-\frac{1}{6}\ln(x^...
35
votes
3answers
2k views

Does the sum of the inverses of the sums of the primes converge?

$$\sum_{m=0}^∞ \frac{1}{\sum_{n=0}^m p_n} = \frac{1}{2} + \frac{1}{5} + \frac{1}{10} + \frac{1}{17} ... $$ Where $p_n$ is the $n$th prime number, does $\sum_{m=0}^∞ \frac{1}{\sum_{n=0}^m p_n}$ ...
35
votes
7answers
3k views

Beautiful identity: $\sum_{k=m}^n (-1)^{k-m} \binom{k}{m} \binom{n}{k} = \delta_{mn}$

Let $m,n\ge 0$ be two integers. Prove that $$\sum_{k=m}^n (-1)^{k-m} \binom{k}{m} \binom{n}{k} = \delta_{mn}$$ where $\delta_{mn}$ stands for the Kronecker's delta (defined by $\delta_{mn} = \begin{...
34
votes
5answers
10k views

How to find the sum of this series : $1+\frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}$

Problem : How to find the sum of this series : $1+\frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}$ This is a Harmonic progression : So is this formula correct to sum the series : $\frac{(...
34
votes
2answers
1k views

Proving $\left(1-\frac13+\frac15-\frac17+\cdots\right)^2=\frac38\left(\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\cdots\right)$

The equality$$\left(1-\frac13+\frac15-\frac17+\cdots\right)^2=\frac38\left(\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\cdots\right)\tag{1}$$follows from the fact that the sum of the first series ...
33
votes
1answer
1k views

Xmas Maths 2014

Inspired the various** algebraic X'mas greetings sent to me over the festive period, I thought I would try to devise one of my own. $$\Large \color{red}{\sum_{i=a-1}^{r-1}}\color{green}{\sum_{j=s-1}^...
32
votes
3answers
2k views

An inequality: $1+\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac53$

$n$ is a positive integer, then $$1+\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac53.$$ please don't refer to the famous $1+\frac1{2^2}+\frac1{3^2}+\dotsb=\frac{\pi^2}6$. I want to find a ...
32
votes
3answers
5k views

Why does Wolframalpha think that this sum converges?

Looking at the sum: $$\sum_{n=1}^\infty\tan\left(\frac\pi{2^n}\right)$$ I'd say that it does not converge, because for $n=1$ the tangent $\tan\left(\frac\pi 2\right)$ should be undefined. But ...
32
votes
4answers
2k views

A closed form of $\sum_{k=0}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$

I am looking for a closed form of the following series \begin{equation} \mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right) \end{equation} I have no idea how to ...
32
votes
4answers
2k views

Questions on a self-made theorem about polynomials

I recently came up with this theorem: For any complex polynomial $P$ degree $n$: $$ \sum\limits_{k=0}^{n+1}(-1)^k\binom{n+1}{k}P(a+kb) = 0\quad \forall a,b \in\mathbb{C}$$ Basically, if $P$ ...
31
votes
12answers
22k views

Why $\sum_{k=1}^{\infty} \frac{k}{2^k} = 2$? [duplicate]

Can you please explain why $$ \sum_{k=1}^{\infty} \dfrac{k}{2^k} = \dfrac{1}{2} +\dfrac{ 2}{4} + \dfrac{3}{8}+ \dfrac{4}{16} +\dfrac{5}{32} + \dots = 2 $$ I know $1 + 2 + 3 + ... + n = \dfrac{n(n+1)}{...
31
votes
1answer
2k views

Witty functional equation

Let $$f(x) = 2/(4^x + 2)$$ for real numbers $x$. Evaluate $$f(1/2001) + f(2/2001) + f(3/2001) + \cdots + f(2001/2001)$$ Any idea?
31
votes
4answers
2k 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 $$f^{(3)}(x)=\sum_{n=0}^{...