# All Questions

17answers
14k views

### How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$

How can I evaluate $$\sum_{n=1}^\infty \frac{2n}{3^{n+1}}$$ I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
17answers
32k views

### How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?

How can one prove the statement $$\lim\limits_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution. This is homework. In my ...
4answers
8k views

7answers
4k views

### Finding the limit of $\frac {n}{\sqrt[n]{n!}}$

I'm trying to find $$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .$$ I tried couple of methods: Stolz, Squeeze, D'Alambert Thanks! Edit: I can't use Stirling.
9answers
3k views

21answers
7k 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 ...
2answers
16k views

Could someone help me prove that given a finite measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$, $\lim_{p\to\infty}\|f\|_p=\|f\|_\... 2answers 4k views ### Prove convergence of the sequence$(z_1+z_2+\cdots + z_n)/n$of Cesaro means Prove that if$\lim_{n \to \infty}z_{n}=Athen: $$\lim_{n \to \infty}\frac{z_{1}+z_{2}+\cdots + z_{n}}{n}=A$$ I was thinking spliting it in: $$(z_{1}+z_{2}+\cdots+z_{N-1})+(z_{N}+z_{N+1}+\cdots+z_{n}... 34answers 26k views ### Why is negative times negative = positive? Someone recently asked me why a negative \times a negative is positive, and why a negative \times a positive is negative, etc. I went ahead and gave them a proof by contradiction like so: Assume ... 3answers 5k views ### How to deduce the CDF of W=I^2R from the PDFs of I and R independent Given pdf of I and R (both I and R are independent RV's), how to find cdf of W =I^2R? Where,$$ \begin{align} f_I(i)&=6i(1-i), &0 \leq i \leq 1 \\ f_R(r)&=2r, &0 \leq r\... 4answers 26k views ### What is 48÷2(9+3)? [duplicate] There is a huge debate on the internet on48÷2(9+3)$. I figured if i wanted to know the answer this is the best place to ask. I believe it is 2 as i believe it is part of the bracket operation in ... 11answers 42k views ### How to prove Euler's formula:$e^{it}=\cos t +i\sin t$? Could you provide a proof of Euler's formula:$e^{it}=\cos t +i\sin t\$ ? thanks.

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