25. Use the integral test in the following analyses: (a) Show that Py n1 n2en converges and...

25. Use the integral test in the following analyses:

(a) Show that Py n¼1 n2en converges and estimate its value. (Hint: integrate by parts.)

(b) Show that Py n¼1 n

n2þ10 diverges, and estimate the rate of growth of the partial sums.

(c) For 0 < q < 1, determine whether Py n¼1 n2qn converges or diverges, and correspondingly estimate its summation value, or the growth rate of its partial sums.

(Hint: Integrate f ðxÞ ¼ x2qx ¼ x2ex ln q using integration by parts.)