Let , expanded about , be used to approximate with a degree- polynomial. For , it can…

MATHEMATICS · Subjective · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Let

f(x)=\ln(1+x)

, expanded about

a=0

, be used to approximate

f(0.6)

with a degree-

n

polynomial. For

0\le x\le0.6

, it can be shown that

|f^{(n+1)}(c)|\le10\,n!

for the relevant

c

. Determine the smallest integer

n

that guarantees the absolute error is less than

1.0\times10^{-7}

; include a short statement verifying the bound is satisfied.