(5) =D4 arctan 239 1. Derive Machin's identity. [Hint: Put a = arctan(). From the relation tan(x +y) = tan x+tan y obtain 1-tan x tan y 4) = 1 119 and tan(4a tan 2a = , tan 4g = 12 239 120 From the...

Number 3 a and b

Extracted text: (5) =D4 arctan 239 1. Derive Machin's identity. [Hint: Put a = arctan(). From the relation tan(x +y) = tan x+tan y obtain 1-tan x tan y 4) = 1 119 and tan(4a tan 2a = , tan 4g = 12 239 120 From the numerical evidence 3. (a) 03 + 13 1 13 + 13 0³ + 1³ + 2³ 24 23 + 23 + 23 36 03 + 13 + 2³ + 33 108 33 + 33 + 33 +33 100 03+13 + 2³ + 33 + 43 320 43 + 43 + 43 + 43 +43 he deduce-as did Wallis–the value of the limit 13 + 23 + 3³ +...+ n³ to L = lim of n→∞ n³ + n³ + n³ + · . . +n³


Extracted text: Chapter 8 410 Use Wallis's method of partitioning by "infinitely (b) small rectangles" to find the area under the curve = x'over the interval [0, a]; in integral notation this amounts to calculating x'dx. y = x %3D a Va Vx dx. 4. Given Wallis's value for " x?dx, obtain y = x2 %3D (Vā, a) Vx dx Nā x² dx х Na 1 Gottfried Leibniz: The Calculus Controversy t