December 2020

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... Read More

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... Read More

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 $\left(-r,r\right)$... Read More

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 $\sqrt{2}$... Read More

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}} $.