Advanced book on Mathematics Olympiad

3.1 Sequences and Series 117

348.Letf:[a, b]→[a, b]be an increasing function. Show that there existsξ∈[a, b]

such thatf(ξ)=ξ.

349.For every real numberx 1 construct the sequencex 1 ,x 2 ,x 3 ,...by settingxn+ 1 =

xn(xn+^1 n)for eachn≥1. Prove that there exists exactly one value ofx 1 for which

0 <xn<xn+ 1 <1 for everyn.

350.Given a sequence(an)nsuch that for anyγ>1 the subsequenceaγnconverges

to zero, does it follow that the sequence(an)nitself converges to zero?

351.Letf:( 0 ,∞)→Rbe a continuous function with the property that for anyx>0,

limn→∞f (nx)=0. Prove that limx→∞f(x)=0.

3.1.5 Series..................................................

A series is a sum

∑∞

n= 1

an=a 1 +a 2 +···+an+···.

The first question asked about a series is whether it converges. Convergence can be

decided using Cauchy’s-δcriterion, or by comparing it with another series. For com-

parison, two families of series are most useful:

(i) geometric series

1 +x+x^2 +···+xn+···,

which converge if|x|<1 and diverge otherwise, and

(ii)p-series

1 +

1

2 p

+

1

3 p

+···+

1

np

+···,

which converge ifp>1 and diverge otherwise.

Thep-series corresponding top=1 is the harmonic series. Its truncation to thenth

term approximates lnn. Let us use the harmonic series to answer the following question.

Example.Does the series

∑∞

n= 1

|sinn|

n converge?

Solution.The inequality|sinx|>

√

2 −√ 2

2 holds if and only if

1

8 <{

x

π}<

7

8 , where{x}

denotes the fractional part ofx(x−x). Because^14 <^1 π, it follows that for anyn, either

|sinn|or|sin(n+ 1 )|is greater than

√

2 −

√

2

2. Therefore,