Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/259


 * (b) Using only the first three terms,


 * the absolute error is less than $$\tfrac{1}{7}$$; i.e. $$\le \tfrac{1}{5040} (= .000198)$$, and the percentage of error is less than $$\tfrac{1}{40}$$ of 1 per cent.


 * Moreover, the exact value of $$sin 1$$ lies between .8333 and .841666, since for an alternating series $$S_n$$ is alternately greater and less than $$lim_{n \to \infty} S_n$$.

Determine the greatest possible error and percentage of error made in computing the numerical value of each of the following functions from its corresponding series
 * (a) when all terms beyond the second are neglected;
 * (b) when all terms beyond the third are neglected.

II. The computation of $$\pi$$ by series. From Ex. 8, &sect;145, we have

Since this series converges for values of $$x$$ between -1 and +1, we may let $$x = \tfrac{1}{2}$$, giving

or

Evidently we might have used the series of Ex 9, &sect;145, instead. Both of these series converge rather slowly, but there are other series, found by more elaborate methods, by means of which the correct value of $$\pi$$ to a large number of decimal places may be easily calculated.

III. The computation of logarithms by series. Series play a very important role in making the necessary calculations for the construction of logarithmic tables. From Ex. 6, &sect;145, we have