1549901369-Elements_of_Real_Analysis__Denlinger_

8.3 Series with Positive and Negative Terms 485

Theorem 8.3.16 A series converges absolutely if and only if each of its sub-

se'ries converges (absolutely).

Proof. Part 1 ( =>): Suppose I: an converges absolutely, and let I: ank be

a subseries. Then, Vm EN, m::; nrn, so

m nm ex:>

L lank I ::; L lakl ::; L lakl·

k=l k=l k=l

Thus, the sequence of partial sums of the subseries I: lank I is bounded

above, so the subseries I: lank I converges absolutely.

P art 2 (-¢:::): Suppose every subseries of the series I: an converges. Then ,
both the series of positive terms and the series of negative terms of I: an
converge, so both I: a;i and I: a;;: converge. By Theorem 8.3.10, this implies
that I: an converges absolutely. •

EXERCISE SET 8.3

1. Prove the assertions made in Example 8.3.6.


2. Prove Lemma 8.3.9.


3. Prove Theorem 8.3.10 (d).
   111111111111


4. Prove that the series - - -+ - - -+ - - -+ - - -+ - - - + - - - +
   2 2 4 4 4 4 8 8 8 8 8 8
   ~ -~ + J_ - · · · converges conditionally, and find its sum.
   8 8 16
   11111 1 1 1


5. Prove that the series - - - + - - - + - - - + · · · + - - - + · · ·
   2 3 4 9 8 27 2k 3k

converges to ~. Does it converge absolutely or conditionally?

6. Determine whether each of the following alternating series converges ab-
   solutely, converges conditionally, or diverges. In case of convergence, find
   an integer no such t hat n ~no=> IS -Snl < .01.
   oo (-1r+i oo (-1)n+i

(a) I: (b) I: --

n=l ..JnTI n=l 3n + 4

(^00) ( -1) n+^1 sin n
(e) n~l n2
oo (-l)n+ln3
(d) n~l (n + 2)!
( f) f ( -1 r ln n
n=l n