represent $x^3 +2x$ as power series

Question

Well it is pretty wierd for me to see this question, the function already is power series isn't it? Am I missing the purpose of the excersize?

Answer 1

A power series is an infinite series of the form $\sum_{n=0}^\infty a_n(x-x_0)^n$. If for some function $f$ and some set $D$, $$ f(x) = \sum_{n=0}^\infty a_n(x-x_0)^n,\quad x\in D $$ we say that the RHS is the power series of $f$ at $x_0$.

When $f$ is a polynomial function of the form $f(x)=b_0+b_1x+\cdots+b_mx^m$, its power series at $x_0=0$ is $f$ itself.

In general, you will need to find the coefficients $a_n$ when $x_0\ne 0$. But Taylor's theorem tells you that it has something to do with $f^{(n)}(x_0)$.

Answer 2

Considering the derivatives of $x^3+2x$ gives the general power series representation about $x=a$ as $$a^3+2a+(3a^2+2)(x-a)+3a(x-a)^2+(x-a)^3$$