Advanced High-School Mathematics

SECTION 5.2 Numerical Series 277

7. As in the above sequence, LetF 0 , F 1 , F 2 , ... be the terms of the
   Fibonacci sequence. Show that

∑∞

k=0

1

Fk

< α,where α =

1 +

√

5

2

(thegolden ratio). (Hint: show that if k ≥2, thenFk > αk−^1.

and then use the ratio test.)^12

8. Consider the generalized Fibonacci sequence (see Exercise 9 on
   page 106) defined byu 0 =u 1 = 1 and un+2= un+1+un. Show
   that ifa, bare such thatun→0 asn→∞, then

∑∞

n=0

unconverges

and compute this sum in terms ofaandb.

5.2.3 Conditional and absolute convergence; alternating se-
ries

In this subsection we shall consider series of the form

∑∞

n=0

an where

the individual terms an are not necessarily non-negative. We shall

first make the following useful definition. An infinite series

∑∞

n=0

an is

calledabsolutely convergent if the series

∑∞

n=0

|an|converges. This is

important because of the following result.

Theorem. If the series

∑∞

n=0

an is absolutely convergent, then it is con-

vergent.

Proof.Note that we clearly have

0 ≤an+|an|≤ 2 |an|, n= 0, 1 , 2 , ....

Since

∑∞

n=0

2 |an|converges, so does

∑∞

n=0

(an+|an|); call the limitL. There-

fore,

∑∞

n=0

an=L−

∑∞

n=0

|an|, proving that

∑∞

n=0

anconverges, as well.

(^12) The above says, of course, that the infinite series of the reciprocals of the Fibonacci numbers
converges. Its value is known to be an irrational number≈ 3. 35988566 ...