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process91
#1
Aug29-11, 10:53 PM
P: 106
This question is not about how to calculate the logarithm, but rather what method would be expected of someone in 1969.

I am going through Apostol's Calculus, and in section 6.10 Apostol introduces polynomial approximations to the natural logarithm. Specifically, he introduces the following theorem:

If [itex]0<x<1[/itex] and if [itex]m \ge 1[/itex] we have

[itex]\ln{\frac{1+x}{1-x}}=2(x+\frac{x^3}{3}+...+\frac{x^{2m-1}}{2m-1})+R_m(x)[/itex]

where the error term, [itex]R_m (x)[/itex], satisfies the inequalities

[itex]\frac{x^{2m+1}}{2m+1}<R_m (x) \le \frac{2-x}{1-x} \frac{x^{2m+1}}{2m+1}[/itex]

The first question of section 6.11 instructs the reader to use this theorem with [itex]x=\frac{1}{3}[/itex] and [itex]m=5[/itex] in order to calculate approvimations to [itex]\ln{2}[/itex]. It instructs to retain nine decimals in the calculations in order to obtain the inequalities [itex]0.6931460<ln{2}<0.6931476[/itex].

Now, I know how to do this; what I mean is that I know the mechanical procedure required to arithmetically calculate this out by hand, but is that really what would be expected of someone in 1969 (when this book was published)? I have tried to do it out by hand three times and failed every time due to a simple mistake at some point. I know calculators were just being developed around that time, but I'm not sure to what extent I should use one, or if I should allow myself to use one at all. What were Apostol's expectations of his students here? I will say, however, that there are 4 questions like this, and given my current rate I may be stuck on these for a while.
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