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.  

Find the Taylor Series expansion of $\cosh x$ at the point $a=0$ and show that the remainder convergences to zero

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

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? $\sum\limits_{n=1}^\infty n! x^n$