# Questions tagged [summation]

1,551 questions
Filter by
Sorted by
Tagged with
355 views

### Find $\sum_{k=1}^{\infty}\frac{1}{2^{k+1}-1}$

Calculate $$\sum_{k=1}^{\infty}\frac{1}{2^{k+1}-1}.$$ I used Wolfram|Alpha to compute it and got it to be approximately equal to $0.6$. How to compute it? Can someone give me a hint or a suggestion ...
31k views

### Calculating the length of the paper on a toilet paper 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 ...
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 ...
2k views

27k views

### Calculate sum of squares of first n odd numbers [closed]

Is there an analytical expression for the summation $$1^2+3^2+5^2+\cdots+(2n-1)^2$$ and how did you derive it?
10k views

93k views

803 views

### How to find answer to the sum of series $\sum_{n=1}^{\infty}\frac{n}{2^n}$

I have put his on wolfram and obtained answer as follows: $\sum_{n=1}^{\infty}\frac{n}{2^n} = 2$ And the series is convergent too because $\lim_{n\to\infty} \frac {n}{2^n} = 0$ However I am ...
902 views

### Summation equation for $2^{x-1}$

Since everyone freaked out, I made the variables are the same. $$\sum_{x=1}^{n} 2^{x-1}$$ I've been trying to find this for a while. I tried the usually geometric equation (Here) but I couldn't get ...
326 views

### How to show that $\sum_{k=1}^n k(n+1-k)=\binom{n+2}3$?

While thinking about another question I found out that this equality might be useful there: $$n\cdot 1 + (n-1)\cdot 2 + \dots + 2\cdot (n-1) + 1\cdot n = \frac{n(n+1)(n+2)}6$$ To rewrite it in a more ...
1k 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 $1/n^2$, $e^{-n}$, ...). ...
9k views

### Inductive Proof for Vandermonde's Identity?

I am reading up on Vandermonde's Identity, and so far I have found proofs for the identity using combinatorics, sets, and other methods. However, I am trying to find a proof that utilizes mathematical ...
377 views

11k views

### Proving $\frac{1}{n+1} + \frac{1}{n+2}+\cdots+\frac{1}{2n} > \frac{13}{24}$ for $n>1,n\in\Bbb N$ by Induction

Proving $\frac{1}{n+1} + \frac{1}{n+2}+\cdots+\frac{1}{2n} > \frac{13}{24}$ for $n>1,n\in\Bbb N$ To solve it I used induction but it is leading me nowhere my attempt was as follows: Lets ...
4k views

### How to derive the formula for the sum of the first $n$ odd numbers: $n^2=\sum_{k=1}^n(2k-1).$ [duplicate]

How to derive this formula? $$n^2=\sum_{k=1}^n(2k-1).$$
13k views

773 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 ...
1k views

### How to prove the inequality $2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$?

Prove that for any positive integer $n$, $$2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$$ Progress I think Riemann sum should be used for ...
852 views

### Proof of the formula $1+x+x^2+x^3+ \cdots +x^n =\frac{x^{n+1}-1}{x-1}$ [duplicate]

Possible Duplicate: Value of $\sum x^n$ Proof to the formula $$1+x+x^2+x^3+\cdots+x^n = \frac{x^{n+1}-1}{x-1}.$$
718 views

3k views

### Can we add an uncountable number of positive elements, and can this sum be finite?

Can we add an uncountable number of positive elements, and can this sum be finite? I always have trouble understanding mathematical operations when dealing with an uncountable number of elements. ...
There is a partial sum formula for $$\sum_{x=1}^n x^1 = \frac{n(n+1)}{2}$$ and even one when the exponent of $x$ is $0$: $$\sum_{x=1}^n x^0 = n$$ but I cannot find one for exponent $-1$: \sum_{x=1}^...
### Proving $\sum_{k=1}^n k k!=(n+1)!-1$
Prove: $\displaystyle\sum_{k=1}^n k k!=(n+1)!-1$ (preferably combinatorially) It's pretty easy to think of a story for the RHS: arrange $n+1$ people in a row and remove the the option of everyone ...