If 0 < a, b < 1, and tan^-1 a + tan^-1 b = π / 4, then the value of (a + b) - [a² + b²] / 2 + [a³ + b³ ] / 3 - [a⁴ + b⁴] / 4 + …. is

a. log_e2

b. log_e (e / 2)

c. e

d. e² – 1

Solution:

Answer: (a)

tan^-1 {[a + b] / [1 – ab]} = (π / 4)

a + b = 1 – ab

(1 + a) (1 + b) = 2

(a + b) – [a² + b²] / 2 + [a³ + b³] / 3 +…infinity

= [a – (a² / 2) + (a³ / 3) …..] + [b – (b² / 2) + (b³ / 3) …..]

= log_e (1 + a) + log_e (1 + b)

= log_e (1 + a) (1 + b) = log_e 2