Combining & Multiplying Power Series

Combining Power Series

Theorem

Suppose that the two power series ${\sum }_{n=0}^{\infty }{c}_n{x}^n$ and ${\sum }_{n=0}^{\infty }{d}_n{x}^n$ converge to the functions $f$ and $g$, respectively, on a common interval $I$.

i. The power series ${\sum }_{n=0}^{\infty }{c}_n{x}^n\pm d_n{x}^n$ converges to $f\pm g$ on $I$.

ii. For any integer $m\ge 0$ and any real number $b$, the power series ${\sum }_{n=0}^{\infty }{b}^m{c}_n{x}^n$ converges to $b^{m}f(x)$ on $I$.

iii. For any integer $m\ge 0$ and any real number $b$, the series ${\sum }_{n=0}^{\infty }{c}_n(b{x}^m{)}^n$ converges to $f(b{x}^m)$ for all $x$ such that $b{x}^m$ is in $I$.

Intuition: If you can represent functions $f$ and $g$ as two power series, then $f\pm g$ is the sum of those two power series. Also multiplying a constant to the power series is equivalent to multiplying the function by it. Pt3 lowkey intuitive already.

Multiplying Power Series

Theorem

Suppose that the power series ${\sum }_{n=0}^{\infty }{c}_n{x}^n$ and ${\sum }_{n=0}^{\infty }{d}_n{x}^n$ converge to the functions $f$ and $g$, respectively, on a common interval $I$. Let

$e_n=c_0{d}_n+c_1{d}_{n-1}+c_2{d}_{n-2}+\dots +c_{n-1}{d}_1+c_n{d}_0={\sum }_{k=0}^n{c}_k{d}_{n-k}$.

Then,

$\left({\sum }_{n=0}^{\infty }{c}_n{x}^n\right)\left({\sum }_{n=0}^{\infty }{d}_n{x}^n\right)={\sum }_{n=0}^{\infty }{e}_n{x}^n$,

and ${\sum }_{n=0}^{\infty }{e}_n{x}^n$ converges to $f(x)\cdot g(x)$ on $I$.

The series ${\sum }_{n=0}^{\infty }{e}_n{x}^n$ is known as the Cauchy product of the series ${\sum }_{n=0}^{\infty }{c}_n{x}^n$ and ${\sum }_{n=0}^{\infty }{d}_n{x}^n$.