For what values of $\alpha$ will $\int_1^\infty \! x^{-\alpha} \, \mathrm{d}x$ converge?

Show that $f(x) = x^4 – 3x^2 + 1$ has two positive and two negative roots.

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.

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

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

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 $x_0=0$.