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

(b) Using only the first three terms,

$\sin 1=1-{\tfrac {1}{6}}+{\tfrac {1}{120}}=.841666,$

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

Moreover, the exact value of $sin1$ lies between .8333 and .841666, since for an alternating series $S_{n}$ is alternately greater and less than $lim_{n\to \infty }S_{n}$.

EXAMPLES

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.

1.	$\cos 1.$	4.	$\arctan 1.$	7.	$e^{-{\tfrac {1}{2}}}.$
2.	$\sin 2.$	5.	$e^{-2}.$	8.	$\arctan 2.$
3.	$\cos {\tfrac {1}{2}}.$	6.	$\sin {\tfrac {\pi }{3}}.$	9.	$\sin 15^{\circ }.$

II. The computation of $\pi$ by series.

From Ex. 8, §145, we have

$\arcsin x=x+{\frac {1\cdot x^{3}}{2\cdot 3}}+{\frac {1\cdot 3x^{5}}{2\cdot 4\cdot 5}}+{\frac {1\cdot 3\cdot 5x^{7}}{2\cdot 4\cdot 6\cdot 7}}+\cdots .$

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

${\frac {\pi }{6}}={\frac {1}{2}}+{\frac {1}{2}}\cdot {\frac {1}{3}}\left({\frac {1}{2}}\right)^{3}+{\frac {1\cdot 3}{2\cdot 4}}\cdot {\frac {1}{5}}\left({\frac {1}{2}}\right)^{5}+\cdots ,$

or

$\pi =3.1415\cdots .$

Evidently we might have used the series of Ex 9, §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, §145, we have

(A)	$\log \left(1+x\right)=x={\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}-{\frac {x^{5}}{5}}-\cdots .$

1. Since $.000198\div .841666=.00023$.