Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am having a hard time understanding how to proceed with this question... Find a power series representation for the function and determine it's radius of convergence

$$ f(x)= x^2\ln(1+x^2) $$

How should I start it out?

share|cite|improve this question
Find the power series for $\ln(1+x^2)$ first. You might find the power series for $1/(1-x)$ helpful. – Nameless Mar 21 '14 at 8:56
This is what confuses me... How do I change that into a form of $$ 1/(1-x) $$ The derivative is $$ 2x/(1+x^2) $$ – StrugglingCalcStudent Mar 21 '14 at 8:57


Take the power series of $\log(1+y)$; replace $y$ by $x^2$; multiply the result by $x^2$.

I am sure that you can take from here.

share|cite|improve this answer
I think I get it... Let me try it out... Thanks – StrugglingCalcStudent Mar 21 '14 at 9:05
@StrugglingCalcStudent. You are welcome and you will get it ! – Claude Leibovici Mar 21 '14 at 9:05

$$1-x^2+x^4-x^6+\cdots=\frac{1}{1+x^2}\tag{by the geometric series}$$


$$x^2-\frac{x^4}{2}+\frac{x^6}{3}-\cdots=\ln(1+x^2)\tag{integrating from $0$ to $x$}$$


Determining radius of convergence is pretty straightforward here.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.