Question

If+y+=+1+++x+++(x^2/2!)+++(x^3/3!)++(x^4/4!)+++.....+infinity,+show+dy/dx=y

Answer 1

Proof :

Given,

y = 1 + x + x²/2! + x³/3! + x⁴/4! + ... + ∞

Differentiating both sides with respect to x, we get

dy/dx = 0 + 1 + (2x)/(2!) + (3x²)/(3!) + (4x³)/(4!) + (5x⁴)/(5!) + ... + ∞

= 1 + (2x)/(2 * 1) + (3x²)/(3 * 2!) + (4x³)/(4 * 3!) + (5x⁴)/(5 * 4!) + ... + ∞

= 1 + x + x²/2! + x³/3! + x⁴/4! + ... + ∞

= y

Thus, dy/dx = y

Hence, proved.