MacLaurin Series Question

[ldizon8]

I don't know how to solve this Maclaurin Series with ease.

Question: Find the Maclaurin series for f(x)=x^(2)e^(-x) . Is this series convergent for x=2? Explain.

I tried getting each single derivative, but it got very messy. Because of that, I couldn't figure out the Maclaurin series for it.

Can someone Please do this problem for me and SHOW EVERY STEP. I would gladly appreciate it, and it would help me out if it was VERY CLEAR, thanks.

 

[stapel]

Can someone Please do this problem for me and SHOW EVERY STEP. I would gladly appreciate it, and it would help me out if it was VERY CLEAR, thanks.

You already have loads of step-by-step clearly-worked-out examples in your textbook, in your class notes, and on the various websites you've consulted in your studies. So one more worked example is unlikely to make much difference. Instead, let's find where things are going wrong.

Question: Find the Maclaurin series for f(x)=x^(2)e^(-x) . Is this series convergent for x=2? Explain.

I tried getting each single derivative, but it got very messy. Because of that, I couldn't figure out the Maclaurin series for it.

Please reply showing your work so far. Once we can see what you're doing, we can provide helpful advice to try to get you back on track. Thank you!

 

[HallsofIvy]

There is no reason to start differentiating \(\displaystyle x^2e^{-x}\). Instead, start with the MacLaurin series for \(\displaystyle e^x\). You probably know that already. If not, it is easy to differentiate! Now, replace every "x" in the series with "-x" to get the MacLaurin series for \(\displaystyle e^{-x}\). Using the fact that \(\displaystyle (-x)^n\) is equal to \(\displaystyle x^n\) if n is even and is equal to \(\displaystyle -x^n\) if x is odd, or that \(\displaystyle (-x)^n= (-1)^nx^n\), should make that easy. Finally, multiply each term in the series by \(\displaystyle x^2\) to get the MacLaurin series for \(\displaystyle x^2e^{-x}\)