Question

[(tanx + tan²x + tan³x + tan⁴x + …. ∞)/πx] for lim x→(0)+ [where 0 < |x| < (π/4)] for

(a) (1/π)

(b) (1/πn)

(c) π

(d) 0

Answer 1

The correct option (a) (1/π)

Explanation:

lim_x→(0)– [(tan x + tan² x + tan³ x + …. ∞)/πx]

∵  a + ar + ar² + ..... ∞ = [a/(1 – r)] ... (sum of infinite series)

∴   lim_x→0 [(tan x)/(1 – tan x)] × (1/πx)

= [lim_x→0 {(tan x)/x}] × (1/π) × lim_x→0 [1/(1 – tan x)]

= 1 × (1/π) × 1

= (1/π)