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

Transcribed Image Text:

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. (b) Integrate from 0 to 1 to obtain 1+x = N Σ(-1)"x" + N In 2 = (-1)³-1 11 In 2-1- +(-1)N+1 (-1)N+1 N+1 1 + x (e) Verify that the integral on the right tends to zero as Noo by showing that it is smaller than fox+1dx. (d) Prove the formula 1 + 2 3 1 1 + x 4 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. (b) Integrate from 0 to 1 to obtain 1+x = N Σ(-1)"x" + N In 2 = (-1)³-1 11 In 2-1- +(-1)N+1 (-1)N+1 N+1 1 + x (e) Verify that the integral on the right tends to zero as Noo by showing that it is smaller than fox+1dx. (d) Prove the formula 1 + 2 3 1 1 + x 4

Answer rating: 100% (QA)

The detailed ... View the full answer