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. loge2

b. loge (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) …..]

= loge (1 + a) + loge (1 + b)

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

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