I have a few doubts about Taylor series expansions, and how you can manipulate to get power series expansion for new functions.
For example I know the power series expansion for $\log(x)$, about $1$ can I get the power series expansion for $\log(x)/x$ by just dividing each term by $x$, I feel like I can do it, but I am not sure if it is justified.
Also I do know that two power series can be multiplied, on their common intersection of radius of convergence, so does it mean we can multiply individual terms of a power series expansion of $f(x)$ by $x^k$ and then obtain the power series for $xf(x)$?
Thank You
Say, $f(x)=\displaystyle\sum_{n=0}^{\infty}a_{n}x^{n+1}$ for $|x|<r$. For each fixed $x\ne 0$, $|x|<r$, the series $\displaystyle\sum_{n=0}^{\infty}a_{n}x^{n+1}$ is about a series of numbers with partial sums $S_{n}(x)=\displaystyle\sum_{k=0}^{n}a_{k}x^{k+1}$ and we have $S_{n}(x)/x=\displaystyle\sum_{k=0}^{n}a_{k}x^{k}$.
Now $S_{n}(x)\rightarrow f(x)$ so $S_{n}(x)/x\rightarrow f(x)/x$ and hence $\displaystyle\sum_{k=0}^{\infty}a_{k}x^{k}$ exists and is $f(x)/x$.
For the multiplication of two power series, consider the partial sums and argue in the similar way.