Multiply each term by 2 cos (the common difference). Thus S= cos a = cos 18 = sin {a+1⁄2 (2n−1) 8} + sin {a + } (2n − 3) 8} sin (a − 18) + (− 1)"−1 sin {a + 1⁄2 (2n − 1) 8}, sin {a + (n − 1) 8} cos and, if n is odd, -= cos (a + d) + cos (a + 28)...+ (− 1)n−1 cos {a + (n − 1) 8}. Then, as in (1), S. cos 18 = cos {a + 1⁄2 (n − 1) 8} cos and, if n is odd, 433. To find = sin {a + 1⁄2 (n − 1) 8} sin and, if n is even. Scosec 2a + cosec 22a + ... + cosec 2o a. We have (see also p. 79) 2S (1 - cos B) − sin a + (n + 1) sin (a + nß) − n sin (a + nß + ß). 436. Similarly, if S= cos(a + B) + 2 cos(a + 2B) + cos a + (n + 1) cos (a + nẞ) − n cos (a + nß + B). Recurring Method. 437. To find the sum of the following series: Let S = sin a + x sin (a + ẞ) + +x+1 sin {a + (r + 1) ß} + Then S. 2x cos ẞ = 2x sin a cos ẞ + + 2x2+1 sin (a + rß) cos ß + And S. x2 = sin a ... ... ... + xn−1 sin {a + (n − 1) B}. + 2x" sin {a + (n − 1) ẞ} cos B. + x2+1 sin {a + (r − 1) ß} + ... + x2 sin {a + (n − 2) ẞ} + x2+1 sin {a + (n − 1) ẞ}. x sin (a - ẞ) - x2 sin (a + nẞ) + x2+1 sin {a + (n − 1) ẞ}. 438. To find the sum of the following series: + 2x2+1 cos (a + rß) cos ß + + 2x2 cos {a + (n − 1) ẞ} cos ß. ... + x1 cos {a + (n − 2) ẞ} + x2+1 cos {a + (n − 1) ẞ}. .. S(1-2x cos ẞ + x2) = cos a − x cos (a - ẞ) − x1 cos (a + nß) + x2+1 cos {a + (n − 1) ß}. cos a cos (a + $) + + cos {a + (n − 1) 6} = 0. ... 5. tan 1⁄2 (n + 1) (# + α) = to n terms COS a- cos 2a + cos 3a Sum to n terms the following series: .... .... .... cos e cos ( + a) + cos (0 + a) cos (0 + 2a) + cos (0 + 2a) cos (0 + 3a) + .... 10. sin 20 cos + sin 30 cos 20+ sin 40 cos 30 + .... 13. sin2 a sin 2a + 1⁄2 sin2 2a sin 4a + sin2 4a sin 8a+ .... 16. sin sin 30 + sin 20 sin 2. 30+ sin 220 sin 22. 30 + 17. cot cox + 2 cot 20 cox 20 + 4 cot 40 cox 40 + 18. sec a sin 2a sec 3a + sec 3a sin 4a sec 5a + .... 19. tan a + cot a + tan 2a + cot 2a + tan 4a + cot 4a+ 20. sin 3a sec2 a sec2 2a + sin 5a sec2 2a sec2 3a + .... .... 24. 23. sin a cos a + sin 3a cos 2a + sin 5a cos 3a + 22 .... sec a + 1 sec a sec 2a + 1 sec a sec 2a sec 4a + 3 cot 3a 3 tan 3a 34. tan & sec 26 + tan 26 sec 46 + tan 40 sec 84 + ... 1 CHAPTER XVIII. ENDLESS SERIES. § 1. CONVERGENCY AND DIVERGENCY. Limits. y 439. IF two quantities x and y vary together in such a way that the difference of x from a may be made less than any assignable value by taking y near enough to b, while the difference of from b may be made less than any assignable value by taking x near enough to a; then we say that in the limit x = a when y=b, or y = b when x = a. If x and y so vary that x may be made greater than any assignable value by taking y near enough to b, while the difference of У from 6 may be made less than any assignable value by taking x large enough; then we say that in the limit x=∞ when yb, or y=b when x = ∞. If x and y so vary that either may be made greater than any assignable value by taking the other large enough; then we say that in the limit x = ∞ when y = or y = ∞ when x = ∞. 440. A sum of an endless number of terms can be used for arithmetical purposes, only if it is convergent. DEF. 1. An endless series is said to be Convergent if there is some one finite quantity towards which the sum of its terms approximates, in such wise that by increasing the number of these terms we may make the difference between their sum and the finite quantity less than any assignable value. And this finite quantity is called the Sum of the endless series. |