486 Chapter 8 n Infinite Series of Real Numbers
- Sum of the Alternating Harmonic Series: Let {Sn} denote the
sequence of partial sums of the alternating harmonic series,
n (-l)k+l 2n 1 n 1
Sn = E. First, show that S2n = E -k - E -k. Then use
k=l k k=l k=l
this, along with the properties of the sequence bn} obtained in Exercise
8.2.41, to show that S 2 n = 12 n - /n + ln 2. Finally, show how this proves
that Sn ____, ln 2. - Prove that (-
11
- · · · + 2-) ____, ln 2 by relating these sums
n+ 1 n+ 2 2n
to the partial sums of the harmonic series and using insights gained from
the previous exercise.
- · · · + 2-) ____, ln 2 by relating these sums
- Use Theorem 8.3. 10 to prove that the alternating series
1 - 1/2 + 1 /2^2 - 1 /3 + 1/3^2 - 1 /4 + 1/4^2 - · · · - l/n + 1/n^2 - · · ·
diverges. Which conditions of the alternating series test do not hold?
10. Determine whether the given series converges absolutely, converges con-
ditionally, or diverges.
(a) f: COS n?T
n=l Vn(c) f; nsin T
n=l 3n + 8( e) f; ( sin n) n
n=l n( )~ sin 7r 2 n
b n=l u n 3/2(d) f; tan(~+¥)
n=l ifTi~ (-l)nsin;;:
(f) 0 [See Example 6.4.7.]
n=l n
(X) xn
11. Prove that for every real number x , the series L I converges absolutely.
n=l n.12. Prove that if E an converges absolutely and {bn} is a bounded sequence,
then L anbn converges absolutely.- Prove that the following series converges conditionally:
1 1 · 3 1 · 3 · 5 1 · 3 · 5 · 7 1 · 3 · 5 · 7 · · · (2n - 1)
---+--- +···+(-l)n+l.
2 2 · 4 2 · 4 · 6 2 · 4 · 6 · 8 2 · 4 · 6 · 8 · · · (2n)
[See Exercises 8.2.43 and 8.2.49; for the sum, see Example 8.7.11.] - Form a rearrangement of the alternating harmonic series by adding the
first two positive terms, then the first two negative terms, then the next
two positive terms, then the next two negative terms, and so on. Prove
that the resulting series converges, and find its sum.
15. Beginning with the harmonic series, form a new series by adding the first
two terms, then subtracting the next two terms, then adding the next two