Mathematical Methods for Physics and Engineering - Matematica.NET

SERIES AND LIMITS4.15 Divide the series into two series, n odd and n even. For r = 2 both are absolutelyconvergent, by comparison with ∑ n −2 .Forr = 1 neither series is convergent,by comparison with ∑ n −1 . However, the sum of the two is convergent, by thealternating sign test or by showing that the terms cancel in pairs.4.17 The first term has value 0.833 and all other terms are positive.4.19 |A| 2 (1 − r) 2 /(1 + r 2 − 2r cos φ).4.21 Use the binomial expansion and collect terms up to x 4 . Integrate both sides ofthe displayed equation. tan x = x + x 3 /3+2x 5 /15 + ···.4.23 For example, P 5 (x) =24x 4 − 72x 2 +9.sinh −1 x = x − x 3 /6+3x 5 /40 − ···.4.25 Set a = D + δ and b = D − δ and use the expansion for ln(1 ± δ/D).4.27 The limit is 0 for m>n,1form = n, and∞ for m