Finding Taylor Series for Exponential Functions

[mmont012]

Hello,

For the exercises in my textbook the directions state:

"Use power series operations to find the Taylor series at x=0 for the functions..."

But now I'm confused; when I see "power series" I think of functions that have x somewhere in them AND there is also the presence of an n.

Here is the first problem in the section:

1. Homework Statement
xe^x

Where do I go from here? There isn't an n in the function at all, so the ratio/root test won't help.

If someone could start me off in the right direction I would much appreciate it! I'm just confused at where to start...

Thank you.

 

[LCKurtz]

Do you know or can you calculate the series expansion of $e^x$? You could multiply it by $x$.

 

[mmont012]

Is that all that you do? I wanted to do that but I thought that it was more complicated... well that is super easy then. Why do they mention power series operations? This was the part that was confusing me the most.

 

[mmont012]

And thank you so much for helping me!

 

[LCKurtz]

Is that all that you do? I wanted to do that but I thought that it was more complicated... well that is super easy then. Why do they mention power series operations? This was the part that was confusing me the most.

Yes, it's a pretty trivial example. A better one would be with a higher power of $x$ like maybe $x^{10}e^x$. That would be just as easy using this method compared to doing its Taylor expansion with all those product derivatives.