Download Solution Manual to Mathematics for Physical Science and by Frank Harris PDF

By Frank Harris

ISBN-10: 0128010002

ISBN-13: 9780128010006

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B) The derivatives of x1/2 at x = a are ( )( ) ( ) dn x1 /2 1 3 1 1 − − ··· − n + 1 a−n+1/2 = dxn x=a 2 2 2 2 = (−1)n−1 (2n − 3)!! (−1)n−1 (2n − 2)! = 2n−1 . an−1/2 These can be inserted into the formula for the Taylor series. (c) Applying the ratio test, we obtain convergence if, for large n, (−1)n (2n − 1)!! (−1)n−1 (2n − 3)!! (x − a)n+1 / (x − a)n < 1 . n+1/2 + 1)! a 2n n! an−1/2 2n+1 (n This condition reduces to |(x − a)/a| < 1, which corresponds to convergence for x within the range (−a, a) and divergence outside that range.

1. (a) Verify the algebraic steps leading to Eq. 88). 1 into your computer and test its function for a wide variety of values of the arguments s, k and P (or p). 1 or by symbolic computation of Zeta( ) or Zeta[ ]. 11. 2. 33 The Euler-Maclaurin integration formula may be used for the evaluation of finite series: [ ] ∫ n n ∑ 1 1 B2 f (m) = f (x) dx + f (1) + f (n) + f ′ (n) − f ′ (1) + · · · . 2 2 2! 1 m=1 Show that (a) n ∑ 1 2 m= n(n + 1). m=1 (b) n ∑ m2 = 1 6 n(n + 1)(2n + 1). m3 = 1 4 n2 (n + 1)2 .

273529E-03 In mathematica, use MakeTable[binom1, 0, 10, 1] to get a similar table after first defining binom1[n_] := Binomial[1/2, n]. 3. Show that for integral n ≥ 0, ∞ ( ) ∑ 1 m m−n = x . n+1 (1 − x) n m=n Solution: Write the binomial expansion (1 − x)−n−1 = ∞ ∑ (−n − 1)(−n − 2) · · · (−n − j) j! j=0 = ∞ ∑ (n + 1)(n + 2) · · · (n + j) j=0 j! (−x)j ) ∞ ( ∑ n+j j x = x . n j=0 j Then change the summation index from j to m = n + j and change the lower limit of the summation from j = 0 to m = n. 4.

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