Given information:

Use Taylor’s Inequality to determine the number of terms of the Maclaurin series for ln(1+x) that should to estimate ln(1.4) to within 0.001

Formula Used:

  Tn(x)→f(x) ???? ??&#??????????;&#??????????;

And Tn(x) can be used as an approximation to

  f:f(x)=Tn(x)

Taylor’s Inequality, which says that if

  |f(n+1)(x)|≤M , then

  |Rn(x)|≤M(n+1)!|x−a|n+1

Calculation:

  f(x)=ln(1+x)

The nth derivative will be

  fn(x)=(−1)n+1(n−1)!(1+x)−n|f(n+1)(x)|≤M(−1)n+2(n)!(1+x)−(n+1)≤Mx=0n!≤Mn!(n+1)!|0.4|n+1≤0.0011(n+1)|0.4|n+1≤0.001

For n=5

  1(5+1)|0.4|6≈0.0068<0.001

  |R3(x)| is less than 0.00001

So, need 5 terms