Calculate $ \lim\limits_{n \to  \infty } \sum \limits_{k=1}^n 4^{-k} $.

Calculate the limit $ \lim\limits_{n \to  \infty } \frac{\left(n^2+3\right)^\frac{1}{2}}{\left(n^2+2\right)^\frac{1}{3}} $.

Calculate the limit $ \lim\limits_{n \to  \infty } \cos \frac{1}{\sqrt{n}}} $.

For what values of $x$ will the following converge absolutely? $\sum\limits_{n=1}^\infty \frac{\left(3x-2\right)^n}{n}$

For what values of $x$ will the following converge absolutely? $\sum\limits_{n=1}^\infty n! x^n$

For what values of $x$ will the following converge absolutely? $\sum\limits_{n=1}^\infty 3^n \frac{ x^n }{ n! } $

For what values of $x$ will the following converge absolutely? $\sum\limits_{n=1}^\infty \left(-1\right)^{n-1} \frac{ x^n }{ n } $

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.