10. Evaluate the convergence or lack of convergence of the following. In the cases of convergence, attempt to determine the formula for NðÞ for arbitrary  > 0, while for divergence toGy, do the same for NðMÞ.

(a) cn ¼

ffiffiffiffiffiffiffiffiffiffiffi n þ 1 mp 

ffiffiffinmp for m A N, m > 1 (Hint: Confirm that am  bm ¼ ða  bÞ

Xm1 j¼0 a jbm1j

!

; ð5:13Þ

and compare to exercise 1(a).)

(b) zj ¼ j!þ j

ð jþ1Þ!

(c) wm ¼ ð1Þmþ1 ln 1 þ 1 m

 

(d) xn ¼ ðn þ 1Þ! þ ð1Þnþ1n!

(e) ak ¼ ð1Þkþ1 2k 10kþk

(f ) bi ¼ ð1Þiþ1ði5  i3 þ 10iÞ

(g) un ¼ ð1Þnþ1np an , p A R, a > 1 (Hint: Consider the value of unþ1 un

:Þ