In this exercise, we show that the Maclaurin expansion of f (x) = ln(1 + x) is valid for
Chapter 10, Problem 86(choose chapter or problem)
In this exercise, we show that the Maclaurin expansion of f (x) = ln(1 + x) is valid for x = 1. (a) Show that for all x = 1, 1 1 + x = N n=0 (1) nxn + (1)N+1xN+1 1 + x (b) Integrate from 0 to 1 to obtain ln 2 = N n=1 (1)n1 n + (1) N+1 1 0 xN+1 dx 1 + x (c) Verify that the integral on the right tends to zero as N by showing that it is smaller than " 1 0 xN+1dx. (d) Prove the formula ln 2 = 1 1 2 + 1 3 1 4 +
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