Let f x = lim n → ∞ ⁡ n n x + n x + n 2 … . . x + n n n ! x 2 + n 2 x 2 + n 2 4 … . . x 2 + n 2 n 2 x n , for all x > 0 . Then

Step-by-step solution

Taking log both side l n f x = lim n → ∞ ⁡ x n l n ∏ r = 1 n x + 1 r n ∏ r = 1 n x 2 + 1 r n 2 1 ∏ r = 1 n r n = x lim n → ∞ ⁡ 1 n ∑ r = 1 n l n x r n + 1 x r n 2 + 1 Express summation in definite integration using r n = t & 1 n = d t = x ∫ 0 1 l n 1 + t x 1 + t 2 x 2 d t Put t x = z l n f x = ∫ 0 x l n 1 + z 1 + z 2 d z ⇒ f ′ x f x = l n 1 + x 1 + x 2 ( ∵ f ( x ) i s a l w a y s p o s i t i v e ) ∀ x ∈ ( 0 , 1 ) ln 1 + x 1 + x 2 > 0 ⇒ f ' ( x ) f ( x ) > 0 ∀ x ∈ ( 0 , 1 ) a s f ( x ) > 0 ⇒ f ' ( x ) > 0 ∀ x ∈ ( 0