Question

How would we integrate over a summation ?

Answer 1

\[\int \left(\sum _{k=0}^{\infty } \frac{x^{k+5}}{k!}\right) \, dx\]

Answer 2

can switch them here right?

Answer 3

can we do it without factoring out e^x

Answer 4

you can integrate each term individually giving:
\[\sum_{0}^{\infty} \int\limits\frac{x^{k+5}}{k!} dx \]

Answer 5

...because integration is a Linear Transformation! *surprise*!!

Answer 6

can change so long as sum converges uniformly. which is basically why "uniform convergence" has a name

Answer 7

so would that work?
\[\sum _{k=0}^{\infty } \frac{x^{k+6}}{k!(k+6)}\]

Answer 8

i believe so.

Answer 9

Thanks