Question

$e^{x^3}$

Short answer

The answer is$e^{x^3}=\sum _{k=0}^{\infty }\frac{x^{3k}}{k!}$

Step-by-step solution

Step 1. Given Information

Consider the function,$f\left(x\right)=e^{x^3}$

Step 2

We know that the Maclaurin series for the function $g\left(x\right)=e^x$is

$e^x=\sum _{k=0}^{\infty }\frac{x^k}{k!}$

So, to find the Maclaurin series for the function $f\left(x\right)=e^{x^3}$we replace $x$by $x^3$in the Maclaurin series of the function $e^x$

Therefore,$e^{x^3}=\sum _{k=0}^{\infty }\frac{{\left(x^3\right)}^k}{k!}$implies thatrole="math" localid="1649860232493" $e^{x^3}=\sum _{k=0}^{\infty }\frac{x^{3k}}{k!}$