Author name: stephen

Prove that polynomials with real coefficients and odd degree always have at least one real root.

Find the Maclaurin Series of $\frac{4x^2-3}{\left(1-x\right)\left(1-2x\right)^2}$ where $x$ is small i.e. less than $\frac{1}{2}$ in magnitude. But, you can’t use

Find a cubic approximation for the $\tan x$ when $x$ is small, i.e. $|x|<\frac{\pi}{2}$.

Find the Taylor Series of $\frac{1}{\left(1-x\right)^2}$

Derive the Taylor Series Expansion (there is no need to find the remainder term / discuss convergence etc.)

Will the following converge or diverge? $\sum\limits_{n=1}^\infty \frac{ x^n }{ n^2 } $

Will the following converge or diverge? $\sum\limits_{n=1}^\infty \frac{ n }{ 3n^2 – 1 } $

Will the following converge or diverge? $\sum\limits_{n=1}^\infty \frac{ n^2 + 1 }{ n^5 + n + 1 } $

Calculate $\sum\limits_{n=1}^\infty \frac{1}{n^2+2n} $.

Calculate $\sum\limits_{n=1}^\infty \frac{1}{4n^2-1} $.