How Do You Find the Maclaurin Series for e^(x^3)?

[NickMusicMan]

I am studying for an exam, and I am trying to figure out:

if you have something like e^(x^3), can you simply substitute x^3 into the M-series for e^x and get the M-series for e^(x^3)? Or would you have to cube the whole e^x series? I have encountered mixed responses to this question.

This is from a practice problem but I am wondering, in general how can you handle ones like f(g(x)), where f has a known expansion.

Thanks,

-NN

 

[Jory]

You can simply substitute x^3 in for x, but remember that the radius of convergence needs to be taken into account. I.e, if f(x) converges for |x| < 2, then f(x^3) would converge for |x^3| < 2.

Try it out though. You can compute the Taylor series from the definition, and then by substituting. They should be equal.

 

[vela]

I am studying for an exam, and I am trying to figure out:

if you have something like e^(x^3), can you simply substitute x^3 into the M-series for e^x and get the M-series for e^(x^3)? Or would you have to cube the whole e^x series? I have encountered mixed responses to this question.

Cubing e^x doesn't give you e^x3. It gives you e^3x.