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Month

November 2020

For what values of $x$ will the following converge absolutely? $\sum\limits_{n=1}^\infty n! x^n$
For what values of $x$ will the following converge absolutely? $\sum\limits_{n=1}^\infty 3^n \frac{ x^n }{ n! }$
For what values of $x$ will the following converge absolutely? $\sum\limits_{n=1}^\infty \left(-1\right)^{n-1} \frac{ x^n }{ n }$
Given that $0 < a < b$ prove $a < \sqrt{ab} < b$ and $\sqrt{ab} < \frac{a+b}{2}$
Prove by induction that $1^3 + 2^3 + \cdots + k^3 = \left(1+2+\cdots+k\right)^2$.
Given that $\sum\limits_{n=1}^\infty b_n$ is a convergent series of positive terms prove that series $\sum\limits_{n=1}^\infty a_n$ of positive terms will...
Show that $\sum\limits_{n=1}^\infty \frac{\sin n}{ n }$ converges.
Prove that the remainder $R_k$ in the Taylor series expansion of $\cos x$ at the point $a=0$ will converge for all values...
Find the limit of $\lim\limits_{n \to \infty } \frac{r^n}{n!}$, where $r$ is a positive real non-zero number.
Prove that $-\pi+\sqrt{5+\pi^2}$ is irrational.