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 $x_0=0$.  

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? $\int_{0}^{3} x^{-\frac{2}{3}} dx$

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

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

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 $[a,b]$.