Advanced book on Mathematics Olympiad

490 Real Analysis

∑

xj< 10 n

1

xj

=

∑n

i= 1

∑

10 i−^1 ≤xj< 10 i

1

xj

<

∑n

i= 1

∑

10 i−^1 ≤xj< 10 i

1

10 i−^1

=

∑n

i= 1

8 · 9 i−^1

10 i−^1

= 80

(

1 −

(

9

10

)n)

.

Lettingn→∞, we obtain the desired inequality.

356.Define the sequence

yn=xn+ 1 +

1

22

+···+

1

(n− 1 )^2

,n≥ 2.

By the hypothesis,(yn)nis a decreasing sequence; hence it has a limit. But

1 +

1

22

+···+

1

(n− 1 )^2

+···

converges to a finite limit (which isπ
2
6 as shown by Euler), and therefore

xn=yn− 1 −

1

22

−···−

1

(n− 1 )^2

,n≥ 2 ,

has a limit.
(P.N. de Souza, J.N. Silva,Berkeley Problems in Mathematics, Springer, 2004)

357.We have

sinπ

√

n^2 + 1 =(− 1 )nsinπ(

√

n^2 + 1 −n)=(− 1 )nsin

π

√

n^2 + 1 +n

.

Clearly, the sequencexn=√ π
n^2 + 1 +n
lies entirely in the interval( 0 ,π 2 ), is decreasing, and

converges to zero. It follows that sinxnis positive, decreasing, and converges to zero.
By Riemann’s convergence criterion,

∑

k≥ 1 (−^1 )

nsinxn, which is the series in question,

is convergent.
(Gh. Sire ̧tchi,Calcul Diferential ̧si Integral(Differential and Integral Calculus),
Editura ̧Stiin ̧tifica ̧ ̆si Enciclopedic ̆a, 1985)

358.(a) We claim that the answer to the first question is yes. We construct the sequences
(an)nand(bn)ninductively, in a way inspired by the proof that the harmonic series
diverges. At step 1, leta 1 =1,b 1 =^12. Then at step 2, leta 2 =a 3 =^18 andb 2 =b 3 =^12.
In general, at stepkwe already knowa 1 ,a 2 ,...,ankandb 1 ,b 2 ,...,bnkfor some integer
nk. We want to define the next terms. Ifkis even, and if

bnk=

1

2 rk

,