Problem 1: Find a closed form for the power series $\sum_{n=2}^{\infty} \tfrac{x^n}{(n-1)n}$. I already found the first two derivatives, but am unsure where to go from here.
Problem 2: Find the length of the curve given by the equation $y= \int_{ -\pi}^{x}\sqrt{\cos(t)} \mathrm dt$ for x between $-\pi$ and $\pi$. I am using the fundamental theorem of calculus to evaluate the integral, and then I will plug that back into the arc length formula, but I'm stuck on how to evaluate the integral.
Also, apologies in advance for not formatting properly, I didn't know how.
Your double differential should result in a Geometric Series, the closed form of which you should know, so just integrate that twice.
Hint: via a change of bound-variable: $$\sum_{n=2}^\infty x^{n-2} ~=~ \sum_{m=0}^\infty x^m$$
For the second, you need to evaluate the arc length via:
$$s= \int_{-\pi}^{\pi} \sqrt{1-\left(\dfrac{\mathrm d y}{\mathrm d x}\right)^2}\mathrm d x\qquad\text{When }y=\int_{-\pi}^x \sqrt{\cos t~}\mathrm d t$$
You do not need to integrate anything to find $y$; you do not even need to find $y$.
Just determine what $\dfrac{\mathrm d y}{\mathrm d x}$ equals via the fundamental theorem.