Convergence of MacLaurin series of function $f(x)=\frac{x}{(1+x^3)}$.

Question

I need to prove the convergence of this MacLaurin series by using the fact that $|f^{(n)} (x)| \leq kM^n \;\;\;\;\forall x \in (-r,r)$.

I found the MacLaurin series and I found the radius of convergence but how do I find the $n$-th derivative and also how do I find $k$ and $M.$

I have also other examples that I need to solve using this convergence criteria so it would be very nice if somebody would help me out and explain the idea how I can solve this.

Thanks in advance.

Answer 1

It is not true that $|f^{(n)}(x)|<kM^n$ for all $x$ in $(-r,r)$.

In fact, the Maclaurin series is $\displaystyle f(x)=\sum_{n=0}^{\infty}(-1)^nx^{3n+1}$. It is obvious that this series converges for $|x|<1$.

Using the formula $\displaystyle f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n$, you can see that $f^{(3n+1)}(0)=(-1)^n(3n+1)!$.

$(3n+1)!$ grows faster than $kM^{3n+1}$ for any $k,M$.