Error Estimation for Mary Boas Series Problem 1.14.8

[kq6up]

Homework Statement

This is problem 1.14.8 in Mary Boas: Math for Phys. Sci.

Estimate the error if $$f(x)=\sum _{ n=1 }^{ \infty }{ \frac { { x }^{ n } }{ { n }^{ 3 } } }$$ is approximated by the sum of its first three terms
for |x| < 1/2 .

Homework Equations

$$Error\quad <\quad \left| \frac { { a }_{ N+1 }{ x }^{ N+1 } }{ 1-\left| x \right| } \right|$$

The Attempt at a Solution

I got the solution manual answer using x=1/2 (Error < 0.002), but shouldn't x=-1/2 be the same error using the equation above? I must be missing something. The manual gives the error .001 for x<0.

 

[vela]

Hint: If x<0, you have an alternating series on your hands.

 

[kq6up]

Haha! I should have seen that. Then the equation for item #2 becomes irrelevant, and you just use the first neglected term for the error approximation.

Thanks,
Chris