Advanced book on Mathematics Olympiad

470 Real Analysis 318.It is known that lim x→ 0 + xx= 1. Here is a short proof using L’Hôpital’s theorem: lim x→ 0 + xx=lim x→ 0 ...

Real Analysis 471 >(n+ 1 )xn+ 1 + 2 ∑n k= 1 kxk− ∑n k= 1 kxk= n∑+ 1 k= 1 kxk, as desired. Furthermore,x 1 >0 by definition ...

472 Real Analysis For our problem, let>0 be a fixed small positive number. There existsn()such that for any integern≥n(), ...

Real Analysis 473 The last two relations imply nlim→∞(byn−yn+^1 )=nlim→∞ bn+^2 byn+yn+ 1 = b √ b− 1 2 . Here we used the fact th ...

474 Real Analysis ··· 1 n <lnn−ln(n− 1 ), we obtain 1 + 1 2 + 1 3 +···+ 1 n < 1 +lnn< 1 +ln(n+ 1 ). Therefore,an<1, ...

Real Analysis 475 Passing to the limit in the recurrence relation, we obtainL= √ L^2 +L−1, and therefore L^2 =L^2 +L−1. But this ...

476 Real Analysis This proves that the sequences are Cauchy, hence convergent. Because asntends to infinity (an,bn,cn)approac ...

Real Analysis 477 333.Define xn= √ 1 + √ 1 + √ 1 +···+ √ 1 ,n≥ 1 , where in this expression there arensquare roots. Note thatxn+ ...

478 Real Analysis computations, that the arithmetic–geometric mean is related to elliptic integrals. The relation that he discov ...

Real Analysis 479 xn+ 1 −( 1 + √ 2 )= √ 1 + 2 xn− √ 1 + 2 ( 1 + √ 2 )= 2 (xn−( 1 − √ 2 )) √ 1 + 2 xn+ √ 1 + 2 ( 1 + √ 2 ) < x ...

480 Real Analysis withx 1 = √ 7 andx 2 = √ 7 − √ Let us first determine the possible values of the limit L, assuming that it ex ...

Real Analysis 481 The conclusion follows. 341.In view of the Cesàro–Stolz theorem, it suffices to prove the existence of and to ...

482 Real Analysis then this will equal the square of the limit under discussion. We use the Cesàro–Stolz theorem. Suppose 0<x ...

Real Analysis 483 = 24 / 4! 22 / 2! = 1 3 . We conclude that the original limit is √ 3. (J. Dieudonné,Infinitesimal Calculus, He ...

484 Real Analysis Now we turn to the geometric mean. Applying the Cesàro–Stolz theorem to the sequences un=ln P( 1 ) 1 m +ln P( ...

Real Analysis 485 Let us write lim n→∞ ank+^1 nk = ⎛ ⎝lim n→∞ a k+k 1 n n ⎞ ⎠ k . Using the Cesàro–Stolz theorem, we have lim n→ ...

486 Real Analysis an+ 1 =an and bn+ 1 = an+bn 2 iff ( an+bn 2 ) < an+bn 2 , or an+ 1 = an+bn 2 and bn+ 1 =bn iff ( an+bn 2 ) ...

Real Analysis 487 Pn′′+ 1 (x)=Pn′′(x) ( 2 Pn(x)+ 1 n ) +(Pn′(x))^2 , andPn(x)≥0, forx≥0. Convexity implies Pn(x)≤ Pn(bn)−P( 0 ) ...

488 Real Analysis To conclude the solution to the problem, assume that the sequence(an)ndoes not converge to 0. Then it has some ...

Real Analysis 489 = 1 x− 1 + 2 n+^2 1 −x^2 n+^2 . This completes the inductive proof. Because 1 x− 1 +nlim→∞ 2 n+^1 1 −x^2 n+^1 ...