Homework SourseI need help with the first part of the problem Let Tnx be th
I need help with the first part of the problem.
Let Tn(x) be the Taylor Polynomial for f(x)=lnx at a=1, and c>1.
Show that |lnc-Tn(c)| <or= (|c-1|^(n+1))/(n+1).
Let Tn(x) be the Taylor Polynomial for f(x)=lnx at a=1, and c>1.
Show that |lnc-Tn(c)| <or= (|c-1|^(n+1))/(n+1).
Solution
ln(1+x) = x- x^2/2 + x^3/3 - ..
so, ln(x)= ln(1+(x-1)) = (x-1) - (x-1)^2/ + (x-1)^3/3 - (x-1)^4/4 + .... (Taylor\'s Exapansion)
Tn(c)= (c-1) - (c-1)^2/ + (c-1)^3/3 -.... + (-1)^n (c-1)^n/n (for |c-1|<1)
|ln(c)- Tn(c)| = |(-1)^(n+1) (c-1)^(n+1)/(n+1) - (-1)^(n+2) (c-1)^(n+2)/(n+2) + (-1)^(n+3) (c-1)^(n+3)/(n+3)... |
<= (c-1)^(n+1)/(n+1)
because |c-1|<1, the terms are decreasing for higher values.
