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 in the set $\left(a,b\right)$ where $a < b$.

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

Prove that polynomials with real coefficients and odd degree always have at least one real root.

Find the Maclaurin Series of $\frac{4x^2-3}{\left(1-x\right)\left(1-2x\right)^2}$ where $x$ is small i.e. less than $\frac{1}{2}$ in magnitude. But, you can’t use the formula for the Maclaurin Series however you may use the fact that $\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots $.

Find a cubic approximation for the $\tan x$ when $x$ is small, i.e. $|x|<\frac{\pi}{2}$.

Find the Taylor Series of $\frac{1}{\left(1-x\right)^2}$

Derive the Taylor Series Expansion (there is no need to find the remainder term / discuss convergence etc.)

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

Will the following converge or diverge? $\sum\limits_{n=1}^\infty \frac{ n }{ 3n^2 – 1 } $