I was helping someone with calc 2 homework (something I haven't done in a while) and need some help with a concept I had trouble explaining. These are the two questions I had trouble with:
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Using the power series for $\cfrac{1}{4-x^2}$ (which is $\sum\cfrac{x^{2n}}{4^{n+1}}$), find the power series for $\cfrac{x^3}{(4-x^2)^2}$.
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Using the power series for $\cfrac{x^3}{1+3x^4}$ (which is $\sum x^3(-3x^4)^n$), find the power series for $\ln(1+3x^4)$.
The solutions given did the following:
For 1. they set the power series equal to $\cfrac{1}{4-x^2}$ and differentiated each side, then doing simple arithmetic to get our wanted power series.
For 2. they took the derivative of $\ln(1+3x^4)$, which gives $12\cfrac{x^3}{1+3x^4}$. We then take the antiderivative to find the wanted power series.
My questions are, why do we have to use the antiderivative for one and not the other, and what's the intuition behind finding the power series through these derivatives? I've never done a problem where you have to find a power series using another power series like this, so I only minimally understand why the solution does what it does.
Thank you!
Best Answer
It's different for every case, but essentially your use of (anti)derivatives comes down to manipulation of the terms of the series.
For example if you knew what $$\sum a_nx^n$$ was and you wanted to know what $$\sum \frac{a_nx^{n+1}}{n+1}$$ was then of course you'd integrate both sides.
Similarly, if you knew what $$\sum \frac{a_nx^{n+1}}{n+1}$$ was and you want $$\sum a_nx^n$$ then you differentiate.
Another example:
We know that $$\frac1{1+x^2}=\sum_{n\geq0}x^{2n}$$ so if we integrate both sides we get $$\arctan x=\sum_{n\geq0}\frac{x^{2n+1}}{2n+1}$$
In short it's just a matter of what you want vs. what you have and doing what you can to turn what you have into what you want.