Question

Let Tn(x) be the Taylor Polynomial for f(x) = lnx at a=1 and c>1........

Answer 1

Show that the max of \[f ^{(k+1)}(x) on [1,c] is f^{(k+1)}(1)\]

Answer 2

Prove \[\left| T _{n}(c)-\ln c \right|\le \left| (c-1 \right|^ {n+1})/(n+1)\]

Answer 3

and Find n such that \[\left| T _{n}(1.5)-\ln 1.5 \right| \le 10^{-2}\]

Answer 4

the first one is easy the f(k+1) derivative will be

Answer 5

\[A*(1/x)^{k+1}\]
This is a decreaing function, so the maximum on \[\left[ 1,c \right]\] will be1.