Problems Integrating an Infinite Series From E&M

[JKreutz]

Hi, guys. For an E&M/quantum mechanics problem I have to integrate the series below:

Homework Statement

Integrate

$\int{r^{2}e^{y r} \, dr}.$2. The attempt at a solution

Using integration by parts and and "differentiating under the integral" give the same answer:

$I= \frac{2 e^{yr}}{y^{3}}-\frac{2 r e^{yr}}{y^{2}}+\frac{r^{2} e^{yr}}{y} + C$,

where I is the solution to the integral and C is an arbitrary constant. I'd read about Newton trying to integrate infinite series, so I thought as a check and out of curiosity I'd express the integrand as an infinite series and integrate each term. Specifically I expressed exp(yr) as a Maclauren series, multiplied each term by the $r^{2}$ and used the "power rule" to get

$I=\sum_{n=0}^{\infty} {\frac{y^{n} r^{(n+2)}}{n! \, (n+3)}}.$

I can't see how this series could represent the other answer I found. Moreover, I've never seen this done before and I don't know to look up this kind of information. (Googling "Infinite Series and Integrals" was not helpful ><). Any help would be appreciated.

 

[hunt_mat]

I wonder if you did it right, if you set r=0 then you should get the answer y^-3, but looking at the series, I don't think you do.

 

[ehild]

Specifically I expressed exp(yr) as a Maclauren series, multiplied each term by the $r^{2}$ and used the "power rule" to get

$I=\sum_{n=0}^{\infty} {\frac{y^{n} r^{(n+2)}}{n! \, (n+3)}}.$

There is a mistake, the power of r has to be n+3. And you need to add a constant to the power series integral, too. Replace exp(yr) in the first integral by its power series. Supposed you are in the range of convergence, the power-series solution is equivalent to the first integral if you match the constants. Find the coefficients of r, r², r³ and so on and compare them.

ehild