Advanced book on Mathematics Olympiad

3.1 Sequences and Series 107 = ( nlim→∞ n+ 1 √nn! )limn→∞nn+ 1 =e. Taking thenth root and passing to the limit, we obtain nlim→∞ ...

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& ...

3.1 Sequences and Series 109 Weierstrass’ theorem.A monotonic bounded sequence of real numbers is convergent. Below are some ins ...

110 3 Real Analysis 329.Show that if the series ∑ anconverges, where(an)nis a decreasing sequence, then limn→∞nan=0. The followi ...

3.1 Sequences and Series 111 3.1.4 More About Limits of Sequences We continue our discussion about limits of sequences with thre ...

112 3 Real Analysis Next, we present a famous identity of S.A. Ramanujan. Example.Prove that √ 1 + 2 √ 1 + 3 √ 1 + 4 √ 1 +··· = ...

3.1 Sequences and Series 113 1 √ 2 (x+ 1 )≤f(x)≤ √ 2 (x+ 1 ). Repeating successively the argument, we find that 2 − 21 n (x+ 1 ) ...

114 3 Real Analysis 339.Show that the sequence √ 7 , √ 7 − √ 7 , √ 7 − √ 7 + √ 7 , √ 7 − √ 7 + √ 7 − √ 7 ,... converges, and eva ...

3.1 Sequences and Series 115 L−  2 + ( −L yn 0 ym +  2 · yn 0 ym + xn 0 ym ) < xm ym <L+  2 + ( −L yn 0 ym −  2 · yn 0 ...

116 3 Real Analysis 347.Letkbe an integer greater than 1. Supposea 0 >0, and define an+ 1 =an+ 1 √kan forn>0. Evaluate nli ...

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. 34 ...

118 3 Real Analysis |sinn| n + |sin(n+ 1 )| n+ 1 ≥ √ 2 − √ 2 2 · 1 n+ 1 . Adding up these inequalities for all odd numbersn, we ...

3.1 Sequences and Series 119 Clearly, the left-hand side of this equality is an integer. For the right-hand side, we have 0 < ...

120 3 Real Analysis 359.Given a sequence(xn)nwithx 1 ∈( 0 , 1 )andxn+ 1 =xn−nxn^2 forn≥1, prove that the series ∑∞ n= 1 xnis con ...

3.1 Sequences and Series 121 ∫b a f(t)dt=F(b)−F(a), whereF′(t)=f(t), becomes the telescopic method for summing a series ∑n 1 ak= ...

122 3 Real Analysis 1 ak+ 1 = 1 ak− 1 − 1 ak+ 1 − 1 , fork≥ 1. Summing up these equalities fork= 1 , 2 ,...,nyields 1 a 1 + 1 +· ...

3.1 Sequences and Series 123 367.For a nonnegative integerk, defineSk(n)= 1 k+ 2 k+···+nk. Prove that 1 + ∑r−^1 k= 0 ( r k ) Sk( ...

124 3 Real Analysis 375.Leta 0 =1994 andan+ 1 = a n^2 an+ 1 for each nonnegative integern. Prove that for 0 ≤n≤998, the number 1 ...

3.2 Continuity, Derivatives, and Integrals 125 Solution.Recall that the Fibonacci numbers satisfy the Cassini identity Fn+ 1 Fn− ...

126 3 Real Analysis Definition.Forx 0 an accumulation point of the domain of a functionf, we say that limx→x 0 f(x)=Lif for ever ...