# Tagged Questions

56 views

215 views

### How to prove $\displaystyle\sum_{n=0}^\infty \frac1{n!}=e\$?

How to prove $\displaystyle\sum_{n=0}^\infty \frac1{n!}=e\$? I thought about it but I could not find a proof. Please give me some hints?
124 views

### Calculate the value of $\sum\limits _{n=1}^{\infty }\:\dfrac{n}{2^n}$ [closed]

In a previous question it is asked to represent $f(x)=\dfrac{x}{1-x^2}$ as a power series. It gave me $\displaystyle\sum _{n=1}^{\infty \:}x\left(2x^2-x^4\right)^{n-1}$. Then they ask to use the last ...
77 views

### How is $2\sum_{n=2}^{\infty}\frac{1}{(n-1)(n+1)}=\frac{6}{4}$ calculated?

$$2\sum_{n=2}^{\infty}\frac{1}{(n-1)(n+1)}=\frac{6}{4}$$ I cant figure out why this is $\frac64$. I try to use telescopic series without success.
493 views

### Prove infinite series

$$\frac{1}{x}+\frac{2}{x^2} + \frac{3}{x^3} + \frac{4}{x^4} + \cdots =\frac{x}{(x-1)^2}$$ I can feel it. I can't prove it. I have tested it, and it seems to work. Domain-wise, I think it might be ...
96 views

### Sum the following $\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)}$

Evaluate: $$\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)}$$ I rewrote the sum as $$\sum_{n=0}^{\infty} \frac {1}{4^{8n-7}(8n-7)} - \sum_{n=0}^{\infty} \frac {1}{4^{8n-3}(8n-3)}$$ Now, I ...
34 views

### Find the sequence $\{c_n\}$ for $c_n = \alpha \cdot c_{n-1} + {\alpha}^{\beta-n}$

Let $\alpha$ and $\beta$ be two given constants, how to find the sequence $\{c_n\}$ for $c_n = \alpha \cdot c_{n-1} + {\alpha}^{\beta-n}$, where $c_0 = {\alpha}^{\beta}$.
40 views

### Confusion Over Sum of Geometric Series

On pg. 88 of A First Course in Probability, it says $$P_i - P_1 = P_1[(q/p) + (q/p)^2 + \cdots + (q/p)^{i-1})]$$ Therefore: $$P_i = \frac{1 - (q/p)^i}{1 - q/p}P_1$$ The series on the right in ...
39 views

### Generalisation of alternating functions

So if we want to have a function go positive negative we take $(-1)^n$, if we want it to take positive positive negative negative(like was on stack exchange a few days ago, we take: ...
67 views

### What is the sum of the power series below?

For $$\sum_{n=1}^{\infty}\frac{(n+2)}{n(n+1)}x^n$$ What is the sum of it?
44 views

### How to prove that $\sum_{n=0}^\infty \frac{1}{(2n+1)^2} + \sum_{k=1}^\infty \frac{1}{(2k)^2}=\frac{4}{3} \sum_{n=0}^\infty \frac{1}{(2n+1)^2}$

How to prove $$\sum_{n=0}^\infty \frac{1}{(2n+1)^2} + \sum_{k=1}^\infty \frac{1}{(2k)^2}=\frac{4}{3} \sum_{n=0}^\infty \frac{1}{(2n+1)^2}$$
75 views