December 2020

Does the following converge or diverge? $\int_{1}^{\infty} x^{-\frac{3}{2}} dx$

Prove that the real numbers are not countable.

Prove that if a function $f$ is bounded and monotonically decreasing on $[a,b]$ then it is Riemann-integrable on $[a,b]$.

Prove $f(x)$ is constant if $|f(x)-f(y)| < (x-y)^2$ for all real $x$ and $y$.

Find the limit $ \lim\limits_{x \to  0 } \frac{ \sin x – x \cos x}{ x^3 } $

Find the limit $ \lim\limits_{x \to  0 } \frac{ \tan x – x}{ x^3 } $

Will the following converge or diverge? $\sum\limits_{n=1}^\infty \frac{ (-1)^n x^n }{ \sqrt{1+n} } $

Will the following converge or diverge? $\sum\limits_{n=1}^\infty \frac{ 2^n x^n }{ n+1 } $  

Prove that p-series test from the integral test.

The functions $h(x)$ and $g(x)$ are differentiable such that the derivative of $h$ is $h$ and the derivative of $g$ is $g$.... Read More