# Part a Using Maple find and show the interval and radius of convergence of this series `sum_(k=0)^oox^(k+1)/(k!)` use ratio test and test the endpoints. Part b to what function does this series...

Part a Using Maple find and show the interval and radius of convergence of this series `sum_(k=0)^oox^(k+1)/(k!)` use ratio test and test the endpoints.

Part b to what function does this series converge. `xe^x` is apparently correct but show why.

*print*Print*list*Cite

**a**. For the Ratio Test, we need to examine the ratio of (k+1)-th coefficient to k-th coefficient, here it is

`(1/((k+1)!)) /(1/(k!)) = (k!)/((k+1)!) = 1/(k+1).`

The limit of this ratio is 0, therefore the power series converges everywhere (and there are no endpoints to check).

**b**. To determine the function to which the series converges, recall the definition of the Taylor series (with the center at `x=0` ). For a function `f(x)` its Taylor series is `sum_(k=0)^oo f^(k)(0) x^k/k!`

Our series is `sum_(k=0)^oo x^(k+1)/(k!) = x sum_(k=0)^oo x^k/(k!) = x e^x,`

because `(e^x)^((k)) = e^x` and `(e^x)^((k))(0) = 1.`