Mathematical Methods for Physics and Engineering - Matematica.NET

SERIES AND LIMITS4.15 Prove that∞∑n=2[ ]n r +(−1) nlnn ris absolutely convergent for r = 2, but only conditionally convergent for r =1.4.16 An extension to the proof of the integral test (subsection 4.3.2) shows that, if f(x)is positive, continuous and monotonically decreasing, for x ≥ 1, and the seriesf(1) + f(2) + ··· is convergent, then its sum does not exceed f(1) + L, whereLis the integral∫ ∞1f(x) dx.Use this result to show that the sum ζ(p) of the Riemann zeta series ∑ n −p ,withp>1, is not greater than p/(p − 1).4.17 Demonstrate that rearranging the order of its terms can make a conditionally∑ convergent series converge to a different limit by considering the series(−1) n+1 n −1 =ln2=0.693. Rearrange the series asS = 1 + 1 − 1 + 1 + 1 − 1 + 1 + 1 − 1 + 1 + ···1 3 2 5 7 4 9 11 6 13and group each set of three successive terms. Show that the series can then bewritten∞∑m=18m − 32m(4m − 3)(4m − 1) ,which is convergent (by comparison with ∑ n −2 ) and contains only positiveterms. Evaluate the first of these and hence deduce that S is not equal to ln 2.4.18 Illustrate result (iv) of section 4.4, concerning Cauchy products, by consideringthedoublesummationS =∞∑n∑n=1 r=11r 2 (n +1− r) 3 .By examining the points in the nr-plane over which the double summation is tobe carried out, show that S can be written asS =∞∑∞∑n=r r=11r 2 (n +1− r) 3 .Deduce that S ≤ 3.4.19 A Fabry–Pérot interferometer consists of two parallel heavily silvered glass plates;light enters normally to the plates, and undergoes repeated reflections betweenthem, with a small transmitted fraction emerging at each reflection. Find theintensity of the emerging wave, |B| 2 ,wherewith r and φ real.B = A(1 − r)146∞∑r n e inφ ,n=0