476 Chapter 8 • Infinite Series of Real Numbers
~1·3·5· .... (2k-1)
- Determine whether the series L.....,,
4
.
6
.
8
..... ( 2 k + 2 ) converges.
k=l
~[ 2·4·6·····(2k) JP
- Use Raabe's test to find values of p for which L.....,, 1.
3
.
5
..... ( 2 k + 1 )
k=l
converges and values of p for which it diverges. - For a, b, c > 0, use Raabe's test to prove that the hypergeometric series
ab a(a + l)b(b + 1) a(a + l)(a + 2)b(b + l)(b + 2)
1+-+ + +·· ·
l!c 2!c(c + 1) 3!c(c + l)(c + 2)converges when c >a+ band diverges when c <a+ b. (Assume c is not
0 or a negative integer.)* 47. Comparison of L:ak and I: ln(l - ak): Suppose that \:/k EN, 0 <
ak < 1. Prove that(a) ak -t 0 ~ ln(l-ak) -t O;
(b) E ak converges~ E ln(l-ak) converges. [Use the limit comparison
test, keeping L'Hopital's rule in mind.]
(c) l:ak diverges~ lim (1-a 1 )(1-a2) · · · (1-ak) = 0. [Apply (b).]
k->oo* 48. A sometimes useful test to prove ak - 0: Suppose { ak} is monotone
decreasing and let bk = 1 - ak+l. Prove that ak -t 0 ~ E bk = +oo.
ak
[Apply Exercise 47 .]*49. 1. 3. 5. 7 ..... (2k - 1)
Apply Exercise 48 to prove that
2
.
4
.
6
.
8
..... ( 2 k) -t 0. [Compare
with Exercise 43 .J8.3 Series with Positive and Negative Terms
There is only one way a nonnegative series can diverge, and that is to +oo.
Similarly, the only way a nonpositive series can diverge is to -oo. As we have
seen in Section 8.2, convergence of such a series is equivalent to boundedness
of its sequence of partial sums. In this section we shall consider convergence
of more general series, whose terms may be positive, negative, or zero. As we
shall see, the convergence behavior of such series can be much more complicated
than that of nonnegative series.