
taylor poly help
Find taylor poly of f(x)=ln(1+x) of order 2 about x=0
done !
found it P2(x)= x
when they say order that means terms right! i,e p2(x)=f(0)+f'(0)x
now the next part which is abit tricky
Find the remainder R2(x)at the point x=1
and use it to prove 13/24< ln2 <5/6
Thanks

Not quite  Order refers to the degree of the Taylor Polynomial.
You want the quadratic p2(x) = xx^2/2
R2(x) = f ''' (t) x^3/3! 0 < t < 1
f ''' = 2/(1+x^3) 0n [0,1] f ''' (x) < 2 anf f'''(x) > 1/4
R2(1) = f ''' (t)/3! < 2/6 =1/3
p2(1) = 11/2 =1/2
ln(2) < 1/2 + 1/3 = 5/6
f ''' (t) > 1/(4*3!) = 1/24
1/2 +1/24 < ln(2)
13/24 < ln(2)