Author name: stephen

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

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

Does the following converge or diverge? $\int_{0}^{1} \ln x dx$

Does the following converge or diverge? $\int_{0}^{1} x^{-1} dx$

Does the following converge or diverge? $\int_{0}^{1} x^{-\frac{1}{2}} dx$

With the help of Rolle’s Theorem and the function $f\left(x\right)=e^{-x} \left(x-a\right)\left(x-b\right)$ prove that the equation $\left(x-a\right)\left(x-b\right)=\left(x-a\right)+\left(x-b\right)$ will have a solution

Does the following converge or diverge? $\int_{0}^{\infty} e^{-x} dx$