1549901369-Elements_of_Real_Analysis__Denlinger_

8.2 Nonnegative Series 473

To find lim k (1 -ak+l) = lim k [1 -(k kP ) ] , we use L'Hopital's

k-+oo ak k-+oo + 1 p

rule:

1 imx. [ 1 - xP ] =1mx 1. [(x+l)P-xP]

X-+00 (x + l)P X-+00 (x + l)P

= lim x x · --~

[

(1 + .!f -1 1 l

X-+00 1 ( 1 + ~) p

( 1 + l) p .- 1 1 ( 1 + l) p - 1

= lim x · lim = lim ~~x=o-c----

x-+oo ~ x-+oo ( 1 + ~) p x-+oo ~

-- 11·m P (1 + ~)p-1 (-x-2)

x-+oo -x-2

(using L'Hopital's rule)

(

1 )p-1

= lim p 1 + - = p.

X-+00 X

Thus, Raabe's test tells us that the p-series converges if p > 1 and diverges
if p < 1. It is inconclusive if p = 1. 0

EXERCISE SET 8.2

1. Prove Theorem 8.2.2.


2. Prove Theorem 8.2.6. Also, express this theorem using the language of
   one series "dominating" another.
   In Exercises 3-12, write the given series in the form I: an and use tests
   given in Sections 8.1 and 8.2 to determine whether the series converges
   or diverges.
   1 1 1 1


3. 1 + 22 + 33 + 44 + 55 +...

4.

5.

6.

7.

1 2 3 4

G3 + 5. 7 + 9. 11 + 13. 15 + ...

1 J2 J3 v'4 V5 J6

3+5+7+9+11+13+···

1 J2 J3 v'4 V5

1.3+2.4 +3·5+ 4.5+5.7+···

1 1 1 1 1

--+--+--+--+--+···

~ J2-3 ~ v;r5 ~