1549901369-Elements_of_Real_Analysis__Denlinger_

8.2 Nonnegative Series 459

00 1

9. Prove that L 2
   5 6

converges, and find its sum.

n=l n + n +

10. Prove that if x is not a negative integer, f ( )(

1

)

n=l x + n x + n + 1

11. Use the method of telescoping series to find ~ 2
    1
    .
    n = l Ln +2n

1

l+x

12. Use partial fractions and other results to find ~ ( ~( ).

n=l Lnn+l n+2

13. Prove that any series can be rewritten as a telescoping series.


14. Let {an} be an arbitrary sequence of positive real numbers. Find a formula

for the nth partial sum of the series f ln (~). For what sequences

n=l an+l

{an} does this series co nverge? In the case of convergence, what is the

sum of the given series?

15. Prove Theorem 8.1.10.

16. Prove Theorem 8.1.11, the Cauchy criterion for series.


17. Prove Theorem 8.1. 12.

8.2 Nonnegative Series

In this section we develop tests for convergence of series having all nonnegative
terms. The assumption of nonnegativity greatly facilitates our investigation of
convergence, and leads to techniques t hat are applicable to more general series
as well.

Definition 8.2.1 A series "2:an is said to be a nonnegative series (or a
series of nonnegative terms) if, Vn, an 2:: 0.

Theorem 8.2.2 A nonnegative series converges if and only if its sequence of
partial sums is bounded above.

Proof. Exercise 1. •

00
Theorem 8.2.3 (The Integral Test) Suppose L an is a nonnegative se-
n = no
ries, and suppose f : [no, +oo) --> IR is a continuous, monotone decreasing