Advanced book on Mathematics Olympiad

108 3 Real Analysis

316.Let(an)nbe a sequence of real numbers with the property that for anyn≥2 there
exists an integerk,n 2 ≤k<n, such thatan=a 2 k. Prove that limn→∞an=0.

317.Given two natural numberskandmleta 1 ,a 2 ,...,ak,b 1 ,b 2 ,...,bmbe positive
numbers such that
√na 1 +√na 2 +···+√nak=√nb 1 +√nb 2 +···+√nbm,

for all positive integersn. Prove thatk=manda 1 a 2 ···ak=b 1 b 2 ···bm.

318.Prove that

nlim→∞n^2

∫ (^1) n
0
xx+^1 dx=

1

2

.

319.Letabe a positive real number and(xn)n≥ 1 a sequence of real numbers such that
x 1 =aand

xn+ 1 ≥(n+ 2 )xn−

∑n−^1

k= 1

kxk, for alln≥ 1.

Find the limit of the sequence.

320.Let(xn)n≥ 1 be a sequence of real numbers satisfying

xn+m≤xn+xm,n,m≥ 1.

Show that limn→∞xnnexists and is equal to infn≥ 1 xnn.

321.Compute

lim

n→∞

∑n

k= 1

(

k

n^2

)k

n^2 +^1

.

322.Letbbe an integer greater than 5. For each positive integern, consider the number

xn= (^11) ︸ ︷︷... (^1) ︸
n− 1
(^22) ︸ ︷︷... (^2) ︸
n

5 ,

written in baseb. Prove that the following condition holds if and only ifb=10:

There exists a positive integerMsuch that for any integerngreater than

M, the numberxnis a perfect square.

We exhibit two criteria for proving that a sequence is convergent without actually
computing the limit. The first is due to Karl Weierstrass.