Prove that p-series test from the integral test.

The functions $h(x)$ and $g(x)$ are differentiable such that the derivative of $h$ is $h$ and the derivative of $g$ is $g$. That is $h'(x)=h(x)$ and $g'(x)=g(x)$, also, the function $h(x)$ is non-zero for all $x$. Prove that $g(x) = k h(x)$ for some constant $k$.

Where (if anywhere) is the function $f(x)=x^3-3x^2+3x-1$ decreasing? Provide a sketch.

Prove  $e-1 \leq \int_{0}^{1} \left(1+x\right)^\frac{1}{2} e^x \leq \sqrt{2} \left(e-1\right) $

If the function $f$ is less than the function $g$ over some interval $a$ to $b$ and both functions are Riemann Integrable from $a$ to $b$ then prove that the integration of $f$ is also less than the integration of $g$ over the interval.

If $a_n$ is a bounded monotonically increasing sequence prove then that $b_n = \frac{a_1 + \cdots + a_n}{n}$ is also bounded and monotonically increasing.

Prove that a bounded monotone increasing sequence is convergent.  

Find the Taylor Series expansion of $\cosh x$ at the point $a=0$ and show that the remainder convergences to zero over $\left(-r,r\right)$ regardless the size of $r$.

For what values of rational $p$ and $q$ will $p\sqrt{2}+q\sqrt[3]{3}$ be rational? You may assume $\sqrt[3]{3}$ is irrational and of course $\sqrt{2}$ is irrational too. Recall though that the sum of two irrationals can be either rational or irrational. Hint: remember how to compare surds.