Essentially what the title says. I'm asked to find the Taylor polynomial of degree $n$ for $f(x)=\sinh^2(x)$ about $a=0$.
This is essentially a Maclaurin series.
I could use the fact that I know what the Maclaurin series of $\sinh(x)$ which is $\sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!}$ and then I could expand term by term.
Is there a better way of doing this though?
Best Answer
Note that $$\sinh ^2 x = (\frac {e^x-e^{-x}}{2})^2=$$
$$(1/4)(e^{2x} +e^{-2x} -2)$$
Now use $$e^{2x} = 1+(2x) + (2x)^2/2 + (2x)^3 / {3!}+.....$$ and $$e^{-2x} = 1+(-2x) + (-2x)^2/2 + (-2x)^3 / {3!}+.....$$ to get your result.