Will the following converge or diverge? $\sum\limits_{n=1}^\infty \frac{ (-1)^n x^n }{ \sqrt{1+n} } $

Will the following converge or diverge? $\sum\limits_{n=1}^\infty \frac{ 2^n x^n }{ n+1 } $  

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 $g$

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

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

Prove that a bounded monotone increasing sequence is convergent.