In this exercise, we show that the Maclaurin expansion of (x) = ln(1 + x) is valid

In this exercise, we show that the Maclaurin expansion of ƒ(x) = ln(1 + x) is valid for x = 1.

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(a) Show that for all x # -1, (b) Integrate from 0 to 1 to obtain 1 1 + x In 2 = = N Σ n=1 N Σ(-1)"x" + n=0 (-1)n-1 n In 2 = 1 - + (-1) N+1 1 (c) Verify that the integral on the right tends to zero as N → ∞ by showing that it is smaller than fxN+¹dx. (d) Prove the formula 2 1 (-1)N+1N+1 1 + x - + if 3 xN+1 dx 1 + x 1 4 +...