Let's say we are given a function defined as ${\frac{\ln(1+t)}{1-t}}$. We want to find the series expansion up to ${t^4}$.
Now we know that we have two function within this larger function which are ${\ln(1+t)}$ and ${\frac{1}{1-t}}$. Now the series and their respective expansions are know and defined as:
$${\ln(1+t)=\sum\limits_{n=0}^{\infty}(-1)^n\frac{x^n}{2}=\frac{1}{2}-\frac{x}{2}+\frac{x^2}{2}-…}$$
$${\frac{1}{1-t}=\sum\limits_{n=0}^{\infty}x^n=1+x+x^2…}$$
Now to find the series expansion as far as ${t^4}$, do we just add or multiply the two series?
Best Answer
You are supposed to multiply the series, $${\frac{\ln(1+t)}{1-t}}= \frac{1}{1-t}\cdot\ln(1+t)=\sum_{n=0}^{\infty}t^n\cdot \sum_{n=1}^{\infty}(-1)^{n+1}\frac{t^n}{n}.$$ As regards the coefficients of the new series see the Cauchy product of two power series.
What do you obtain in this case? What is the series expansion up to $t^4$?