- (b) Using only the first three terms,

- the absolute error is less than ; i.e. , and the percentage of error is less than of 1 per cent.
^{[1]}

- Moreover, the
*exact value*of lies between .8333 and .841666, since for an alternating series is alternately greater and less than .

**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. |
4. |
7. |
|||

2. |
5. |
8. |
|||

3. |
6. |
9. |

**II.** *The computation of by series.*

From Ex. 8, §145, we have

Since this series converges for values of between -1 and +1, we may let , giving

or

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 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) |

- ↑ Since .