Prove that $e^a (b-a) < e^b – e^a < e^b(b-a)$ for all $a<b$ in $\mathbb{R}$.

Consider the non-zero real-valued function $exp$ on the reals such that $exp(a+b)=exp(a)exp(b)$ and $exp'(a)=exp(a)$ for all real values of $a$

Where does the following converge $\int_{1}^{\infty} e^{-\alpha x} dx$?

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.