Test for absolute convergence: $\displaystyle\sum_{n=1}^{\infty}\frac{n! (x-2)^n}{ n^2 }$

Test for absolute convergence: $\displaystyle\sum_{n=1}^{\infty}\frac{(x-4)^n}{ n }$

Test for absolute convergence: $\displaystyle\sum_{n=1}^{\infty}\frac{2^n x^n}{ n^2 }$

Consider the sequence $ x_{n} = \frac{1}{n} $ if $n$ is odd and $x_{n} = 1$ if $n$ is even. Does this sequence converge?

Consider the sequence $ x_{n} = \frac{1}{n} $ if $n$ is odd and $x_{n} = 0$ if $n$ is even. Does this sequence converge?

Prove that $\sqrt{2}+\sqrt{3}$ is irrational without assuming any particular surd is irrational i.e. you must prove first that a surd like $\sqrt{6}$ is irrational.

Prove that cubic equations (of real coefficients) must have at least one real root.

Find the region of convergence for the Taylor Series of $\ln(1+x)$ expanded at $x_0=0$.  

Find the region of convergence for the Taylor Series of $\frac{1}{1+x}$ expanded at $x_0=0$.

Does the following converge or diverge? $\int_{0}^{3} x^{-\frac{2}{3}} dx$