458 Chapter 8 • Infinite Series of Real Numbers
EXERCISE SET 8.1- Prove that ~ L.., ---2_ 10n converges, and find its limit. What does this tell you
n=O
about the infinite, nonterminating decimal 0.99999999 · · ·? - Find the sum of each of the following series, if it converges:
(a) 0.0101010101 · · ·
(c) 1 - ~ + i -~ + · · ·
oo 3n + 4n
(e) L 5n
n=O
00
(i) L sin(mr)
n=l(b) 0. 987698769876 ...
(d) ~ - i + 211 - s\ + · · ·
(f) ~ 3n + 5n
L.., 4n
n=O
oo n
(h) n~l 2n + 3(j) n~l sin ( n27r)00 00- Prove that L cos nx diverges Vx E JR, and L sin nx diverges unless x
n=l n=l
is an integral multiple of 7r.
00 00 - Define the m-tail of a series L an to be the series L an. Show that
n=l n=m
Vm E N, a given series converges if and only if its m-tail converges.
oo m-1 oo
Show that in case of convergence, L an = L an+ L an. How does
n=l n=l n=m
the behavior of m-tails of series differ from the behavior of m-tails of
sequences? [See Definition 2.2. 15 and Theorem 2.2.16.] - Prove that altering or deleting a finite number of terms of an infinite
series does not affect its convergence or divergence. - Prove that every sequence {an} is the sequence of partial sums of some
series L Xk·
00 - Prove the claim made in Example 8.1.8: A telescoping series L (bn-bn+i)
n=l
converges if and only if the sequence {bn} converges; in fact, if bn --> B ,
00
then L (bn - bn+1) =bi - B.
n=l
00 1 - Prove that L - 2 --is a telescoping series, and find its sum. [Hint: sep-
n=l n +n
1
arate - 2 --into two fractions, using the method of "partial fractions."]
n +n