Let fx ln1 x x Solutiona For the first part of expansion c

Solution

a) For the first part of expansion, cosider 1/1+x. The Taylor expansion is 1/1+x = 1- x + x² - x³ +...

Integrating both sides, ln(1+x) = x - x²/2 + x³/3 - x⁴/4

Remainder theorem says that f(x) = x - x²/2 + x³/3 - x⁴/4 + r_n(x)

where r(x) < Integral 0 to x (-x^k/1+x) dt = -x^k+1/1+x

Note that r(x) -> 0 as k-> and the stronger bound is given by R_n(x) = ((-1)ⁿ/n+1)(x/(1+z))ⁿ⁺¹

Hence f(x) =x - x²/2 + x³/3 - x⁴/4 + ((-1)ⁿ/n+1)(x/(1+z))ⁿ⁺¹