Use the formula for the sum of a geometric series to find the sum or state that the series diverges (enter DIV for a divergent series). sum_{n=2}^inftyfrac{5^n}{12^n}

Proving identity
${\displaystyle \sum [{(j+t)}^{-1}-j^{-1}]=\sum _{k\ge 1}\zeta (k+1){(-t)}^k}$

Let $f:[0,1]\to \mathbb{R}$ be continuous with $f(1)=f(0)$. Prove that if $h\in (0,\frac{1}{2})$ is not of the form $\frac{1}{n}$, then there does not necessarily exist $|x-y|=h$ satisfying $f(x)=f(y)$. Provide an example that illustrates this using $h=\frac{2}{5}$.
So I was given the hint that I can use a modified $sin$ function, however I'm not really sure how I would go about that. Preferably, an example not using that would be great.
My thoughts so far are that I use a proof by contradiction saying that $\mathrm{\exists }$ $x,y\in [0,1]$ such that $f(x)=f(0)$ and $|x-y|=h$. And I need to get to the point where $h=\frac{1}{n}$, which would create the contradiction (I assume that's the endpoint?). However, how would I use $h=\frac{2}{5}$ to prove that? Is it trivial in that $h=\frac{1}{n}$ is false? Also, it isn't given that $n\in \mathbb{N}$, so maybe I shouldn't assume that?

How do I compute the maximum value of the solution of this differential equation?
$\{\begin{array}{rl}\frac{dy}{dt}& =f(y)=1+kϵ-ky-e^{-y}\\ y(0)& =ϵ\end{array}$

Solving partial differential equation in 2d with 3 boundary conditions
Consider the following pde:
${\displaystyle -\mathrm{\Delta }u=0}$
in ${\displaystyle (0,\pi )\times (0,4)}$ with boundary conditions:
${\displaystyle u(0,y)=u(\pi ,y)\text{for}y\in (0,4)}$
and
${\displaystyle u(x,0)=\cos x\sin x\text{for}x\in (0,\pi )}$
and
${\displaystyle u(x,4)=\cos x\sin x+4\text{for}x\in (0,\pi )}$

How do you find the rectangular coordinates of the point whose spherical coordinates are ${\displaystyle (4,3\frac{\pi }{4},\frac{\pi }{3})?}$

Represent the plane curve by a vector-valued function. (There are many correct answers.)
${\displaystyle 3x+4y-12=0}$

Given the following series: ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}{(\sqrt[3]{n+1}-\sqrt[3]{n-1})}^{\alpha }}$ where ${\displaystyle \alpha \in \mathbb{R}}$. Does the series converge or diverge?