Given that $0 < a < b$ prove $a < \sqrt{ab} < b$ and $ \sqrt{ab} < \frac{a+b}{2}$

Prove by induction that $1^3 + 2^3 + \cdots + k^3 = \left(1+2+\cdots+k\right)^2$.

Given that $\sum\limits_{n=1}^\infty b_n $ is a convergent series of positive terms prove that series $\sum\limits_{n=1}^\infty a_n $ of positive terms will also converge if $\lim\limits_{n \to  \infty } \frac{a_n}{b_n} = L > 0$. Hint: You may use the Direct Comparison Test.

Show that $\sum\limits_{n=1}^\infty \frac{\sin n}{ n } $ converges.

Prove that the remainder $R_k$ in the Taylor series expansion of $\cos x$ at the point $a=0$ will converge for all values of $x$ in $\mathbb{R}$.  

Find the limit of $ \lim\limits_{n \to  \infty } \frac{r^n}{n!} $, where $r$ is a positive real non-zero number.  

Prove that $-\pi+\sqrt{5+\pi^2}$ is irrational.  

Find the limit of $ \lim\limits_{n \to  \infty } \sqrt[n]{n} $.  

Calculate the upper and lower Riemann Sum for the function $f\left(x\right)=1$ when $x$ is rational and $f\left(x\right)=-1$ when $x$ is irrational. Hence show that this function is not Riemann-integrable.

Calculate the limit \lim\limits_{x \to 0 } x \ln x without using L’Hôpital’s rule.