Chapter 1.1, Problem 49E

use Corollary 12.6.2 and 12.6.3 to derive 12.6.4,12.6.5, 12.6.6 and 12.6.7

Explain the focus and reasons for establishment of 12.5.1(lim(n->infinite) and sigma of k=0 to n)

Explain the focus and reasons for establishment of 12.5.3 about alternating series. and explain the reason why (sigma k=1 to infinite)(-1)k+1/k = 1/1 - 1/2 + 1/3 - 1/4 + .... converges.

Explain the key points and reasons for the establishment of 12.3.2(integral Test)

Use identity (1+x+x2+...+xn)*(1-x)=1-xn+1 to derive the result of 12.2.2. Please notice that identity doesn't work when x=1.

Explain the key points and reasons for the establishment of 11.3.2(integral Test)

To explain how to view "Infinite Series" from "Infinite Sequence"’s perspective, refer to 12.2.1

Explain the key points and reasons for the establishment of 12.2.5 and 12.2.6

8. For x>_1, the continuous function g is decreasing and positive. A portion of the graph of g is shown above. For n>_1, the nth term of the series summation from n=1 to infinity a_n is defined by a_n=g(n). If intergral 1 to infinity g(x)dx converges to 8, which of the following could be true? A) summation n=1 to infinity a_n = 6. B) summation n=1 to infinity a_n =8. C) summation n=1 to infinity a_n = 10. D) summation n=1 to infinity a_n diverges.