This is the value of π correct to 8 places of decimals. By taking the first series to 21 terms and the second series to three terms we should get π correct to sixteen places. 349. Rutherford's Series. A further simplification of Machin's formula is the expression Find the value of π to three places of decimals 9. By using Euler's Series. 10. By using Machin's Series. 11. By using Rutherford's Series. 12. To the second order of small quantities, prove that 1 1+sin @ log (1 - 0) + tan-1 0 sin (5 + 0) = √3 - 10. Ꮎ . 2 3 ૭) 13. When both and tan-1 (sec ) lie between 0 and CHAPTER XXVIII SUMMATION OF SERIES. EXPANSIONS IN SERIES. 350. WE shall now apply the results of the preceding chapters to the summation of some trigonometrical series. The chief series may be divided into four classes; (1) Those depending for their summation on a Geometrical Progression ultimately, (2) Those depending ultimately on the Binomial Theorem, (3) Those depending ultimately on the Exponential Theorem, including, as sub-cases, the Sine and Cosine Series, and (4) Those depending ultimately on the Logarithmic Series and, as a sub-case, Gregory's Series. 351. In Arts. 352-355 we shall sum one example of each of these classes. It will generally be found more convenient in summing one of these series involving sines of multiple angles (such as sin a, sin 2a, sin 3a ...) to also sum at the same time the companion series involving the cosines of the same multiple angles (i.e. cos a, cos 2a, cos 3a ...). The method will be best seen by a careful study of the following four articles. 352. Ex. Sum to n terms, and to infinity, the series 1+ c cos a + c2 cos 2a + where c is less than unity. Let C = 1+c cos a + c2 cos 2a + ... + c2−1 cos (n − 1) a ..............(1), and S=c sin a+ c2 sin 2a + .............. + c2¬1 sin (n − 1) a......................... (2). Multiplying (2) by i and adding to (1), we have C + Si = 1+c (cos a + i sin a) + c2 (cos 2a + i sin 2a) + =1+ cear + c2 e2ai + ... + c^-1 e (n−1) ai 1 - cn enai +cn-1 e(n-1) ai (Art. 308) 1- ceai, by summing the G.P., {1 — c2 (cos na + i sin na)} 1- c cos a - ic sin a {1 - c2 cos na (Art. 308) ic2 sin na} {1-c cos a + ic sin a} (1 − c cos a)2 + c2 sin2 a {(1 − c cos a) (1 − c2 cos na) + c2+1 sin na sin a} +i {c sin a (1 − c2 cos na) — c2 sin na (1 c cos a)} Hence, by equating real and imaginary parts, we have C= (1 1 − c cos a) (1 and S= c2 cos na) + c2+1 sin na sin a 1 - 2c cos a + c2 c sin a (1 — c2 cos na) — c2 sin na (1 − c cos a) 1- 2c cos a + c2 c2 cos na + c2+1 cos (n − 1) a 1 2c cos a + c2 c2 sin na + c2+1 sin (n − 1) a 1- 2c cos a + c2 |