How many terms of the Maclaurin series of f(x) = ln(1+x) are needed to compute ln(1.5) to within an error of at most 0.0001 ?

Answer 1

^{First, the Maclaurin series expansion of f(x) = ln(1+x) is}

ln(1+x) = x-x²/2+x³/3-x⁴/4+...

So we need the nth term to be less than 0.0001, and with x=0.5

0.5ⁿ/n ≤0.0001

Note that

0.5⁹/9 ≈ 0.000217

0.5¹⁰/10 ≈ 0.0000976

Since, 0.5¹⁰/10 < 0.0001, we need nine terms to compute ln(1.5) to within 0.0001.