Chapter 5.5, Problem 315E

The following alternating series converge to given multiples of $\pi$ . Find the value of N predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum N for which the error bound holds, and give the desired approximate value in

each case. Up to 15 decimals places,

   $\pi =\mathrm{3.141592653589793...}$

315. [T] The Euler transform rewrites = n=O )“b,, as S= (—1)’2” I (Z1)b_1. For the

n=O

alternating harmonic series, it takes the form

-I

—‘ (—1)” 1 In(2) = = L ,.• Compute partial

n=I n=I

sums of ,, until the’ approximate in(2) accurate

n=I fl2

to within 0.0001. How many terms are needed? Compare this answer to the number of terms of the alternating harmonic series are needed to estimate ln(2).