1549901369-Elements_of_Real_Analysis__Denlinger_

472 Chapter 8 • Infinite Series of Real Numbers

Since the sum on the left is telescoping, this says

Thus, the partial sums of L ak are bounded, so it converges.

(b) Suppose the hypotheses of (b) hold. Then 3 0 < r :::; 1, n 0 E N 3

Thus, { kak+ 1 } is monotone increasing for k 2: n 0. Thus, from these in-

equalities,

So, 3 c > 0 3

c

k 2: no =? ak+I 2: k.

But L ~ diverges, so by the comparison test, L ak diverges. •

Corollary 8.2.21 (Raabe's Test, Limit Form) Suppose 2:ak is a series

of positive terms, and suppose R = lim k (1 -ak+l) exists.

k->oo ak

(a) If R > 1, then L ak converges.

(b) If 0 < R < 1, then L ak diverges.

( c) The test is inconclusive if R = l.

Proof. Exercise 42. •

Example 8.2.22 Both the ratio test and the root test are inconclusive for the
1
p-series L kP (Exercise 35). To apply Raabe's test to the p-series we first note

that

ak+l (k + 1)-p kP

ak k-P (k+l)P.