How many nonzero terms of the maclaurin series for
ln(1 + x) do you need to use to estimate ln 1.4 to within 0.0001?

The Maclaurin series for is



and converges for . To approximate to within 0.0001 is the same as finding such that



where is the th order Taylor approximation to and is the remainder term.

By Taylor's theorem, the remainder term is bounded above by



where is the maximum value achieved by in some interval centered at . We have




Let's consider the interval . The th derivative is maximized at , and this leaves us with



From this we have that



So now we look for such that



Using a calculator, we can deduce that this occurs for all integers , which means we need at least 7 terms of the series to achieve the desired accuracy.

Let's check this result:







Taking the absolute difference, you'll see that , but this is not the case with .