Advanced book on Mathematics Olympiad

3.2 Continuity, Derivatives, and Integrals 147 The generalized mean inequality.Given the positive numbersx 1 ,x 2 ,...,xnand the ...

148 3 Real Analysis We begin by recalling the basic facts about indefinite integrals. Integration is the inverse operation to di ...

3.2 Continuity, Derivatives, and Integrals 149 With the substitutionu=x^1 a+^12 the integral becomes − 1 a ∫ 1 √ u^2 +^34 du=− 1 ...

150 3 Real Analysis 3.2.8 Definite Integrals......................................... Next, definite integrals. Here the limits ...

3.2 Continuity, Derivatives, and Integrals 151 457.Compute the integral ∫a 0 dx x+ √ a^2 −x^2 (a > 0 ). 458.Compute the integ ...

152 3 Real Analysis Proof.To prove the formula we start by computing recursively the integral In= ∫ π 4 0 tan^2 nxdx, n≥ 1. We h ...

3.2 Continuity, Derivatives, and Integrals 153 465.Letn≥0 be an integer. Compute the integral ∫π 0 1 −cosnx 1 −cosx dx. 466.Comp ...

154 3 Real Analysis as 1 n [ 1 1 +^1 n + 1 1 +^2 n +···+ 1 1 +nn ] , we recognize the Riemann sum of the functionf:[ 0 , 1 ]→R,f ...

3.2 Continuity, Derivatives, and Integrals 155 This is just a Riemann sum of the function( 1 − 2 x)ln( 1 −x)over the interval[ 0 ...

156 3 Real Analysis 3.2.10 Inequalities for Integrals................................... A very simple inequality states that if ...

3.2 Continuity, Derivatives, and Integrals 157 475.Letf:[ 0 , 1 ]→Rbe a continuous function such that ∫ 1 0 f(x)dx= ∫ 1 0 xf (x) ...

158 3 Real Analysis As an instructive example we present in detail the proof of another famous inequality. Chebyshev’s inequalit ...

3.2 Continuity, Derivatives, and Integrals 159 480.Letf:[ 0 ,∞)→[ 0 ,∞)be a continuous, strictly increasing function withf( 0 )= ...

160 3 Real Analysis f(x)=f(a)+ f′(a) 1! (x−a)+ f′′(a) 2! (x−a)^2 +···+ f(n)(a) n! (x−a)n+···. Ifa =0, the expansion is also know ...

3.2 Continuity, Derivatives, and Integrals 161 ∫ 1 0 lnxln( 1 −x)dx= ∑∞ n= 1 1 n(n+ 1 )^2 . Using a telescopic sum and the well- ...

162 3 Real Analysis We conclude the list of examples with the proof of Stirling’s formula. Stirling’s formula. n!= √ 2 πn (n e ) ...

3.2 Continuity, Derivatives, and Integrals 163 To bring this closer to Stirling’s formula, note that the term in the middle is e ...

164 3 Real Analysis 487.Compute the ratio 1 +π 4 5 !+ π^8 9! + π^12 13 !+··· 1 3 !+ π^4 7 !+ π^8 11 !+ π^12 15 !+··· . 488.Fora& ...

3.2 Continuity, Derivatives, and Integrals 165 This expansion is unique, and a 0 = 1 2 π ∫T 0 f(x)dx, an= 1 π ∫T 0 f(x)cos 2 nπ ...

166 3 Real Analysis LetCn(x)= ∑∞ n= 1 1 n^2 + 1 cosnxandSn(x)= ∑∞ n= 1 n n^2 + 1 sinnx. They satisfy 1 2 +Cn(x)+Sn(x)= πex e^2 π ...