6. Use the integral test in the following analyses: (a) Show that Py n1 en converges and...

6. Use the integral test in the following analyses:

(a) Show that Py n¼1 en converges and estimate its value. (Note: This can also be summed exactly as a geometric series, of course, but that is not what is to be done here.)

(b) Show that Py n¼1 nm, for mb1, diverges, and estimate the rate of growth of the partial sums.

(c) For 0 < q < 1, determine whether Py n¼1 nqn converges or diverges, and correspondingly estimate its summation value, or the growth rate of its partial sums.
(Hint: Integrate f ðxÞ ¼ xqx ¼ xex ln q using integration by parts.)