$$ \lim_{n\to\infty}\frac n{2^n}=0. $$ I know how to prove it by using the trick, $2^n=(1+1)^n=1+n+\frac{n(n-1)}{2}+\text{...}$ But how to prove it withou...

I don't understand why $2, 147, 483, 647$ is the max number for a $32$-bit integer. $8$ bits $= 1$ byte $32$ bits $= 4$ bytes How is this calculated?...

I am extremely interested in the Putnam competition. Given that I am currently in 11th grade (so I have 12th grade and half my freshman year to study (plu...

I can find this using the fact that $\sin(\sin^{-1}(x)) = x$, for all $x\in[-1,1].$ Now, differentiate. $$\frac{d}{d\sin^{-1}(x)}\sin(\sin^{-1}(x))\cdot \...

Suppose that l,m,k,t differnt lines (Like this illustration) Describe accurately this composition: $S_k\circ\ S_t\circ\ S_m\circ\ S_l$ Well, composition o...

I'm pretty new to matrices, and only know the basics at the moment. I was given a question that asks: For the system of equations: $$ \begin{split} x...

I am doing some work on probability. I have done some background reading on the definitions on exhaustive, non exhaustive and mutually exclusive but the d...

Given Two Fields $F,K$, and two vector spaces $V,W$ over $F$, what does tensor product $$V\otimes_{K} W$$ mean? I am not certain whether this is defined i...

What I know from basic calculus: The first derivative of a function $f:\mathbb{R}\to\mathbb{R}$ gives an indication of the function's behavior (namel...

Using the grade school method of multiplying two binary numbers takes $O(n^2)$ time, where $n$ is the length of the number in bits. Why is this true?