Question

The number of real solutions of the equation sin^-1(∑x^{i + 1} - x∑(x/2)ⁱ for i ∈ [i=1, ∞])

lying in the interval (- 1/2, 1/2) is ........ .

(Here, the inverse trigonometric functions sin^-1x and cos^-1x assume values in (- π/2, π/2), and [0, π] respectively.)

Answer 1

Given:

Differentiating, we get

Thus, f(0) < 0 and f(1/2) > 0.

Hence, there are two roots in interval (-1/2, 1/2).