Evaluate Integral as a power series Question

[grib90]

Evaluate the integral as a power series and find the radius of convergence:

f(x) = [e^(-t^3)]dt

Not many useful examples of these I can find, so any help is appreciated.

 

[pka]

\(\displaystyle e^x = \sum\limits_{k = 0}^\infty {\frac{{x^k }}{{k!}}} \quad \Rightarrow \quad e^{ - t^3 } = \sum\limits_{k = 0}^\infty {\frac{{\left( { - t^3 } \right)^k }}{{k!}}} = \sum\limits_{k = 0}^\infty {\frac{{\left( { - 1} \right)^k t^{3k} }}{{k!}}}\)

 

[grib90]

\(\displaystyle e^x = \sum\limits_{k = 0}^\infty {\frac{{x^k }}{{k!}}} \quad \Rightarrow \quad e^{ - t^3 } = \sum\limits_{k = 0}^\infty {\frac{{\left( { - t^3 } \right)^k }}{{k!}}} = \sum\limits_{k = 0}^\infty {\frac{{\left( { - 1} \right)^k t^{3k} }}{{k!}}}\)

Thanks for the reply. Since it's an integral wouldn't you have to integrate it? But how do you deal with factorials when integrating?

 

[Deleted member 4993]

Thanks for the reply. Since it's an integral wouldn't you have to integrate it? But how do you deal with factorials when integrating?

k is constant - term by term - thus does not affect integration