Question

If `lim_(n rightarrow ∞) (1^a + 2^a + ......... + n^a)/((n + 1)^(a - 1)[(na + 2) + ......(na + n)]) = 1/60` for some positive real number a, then a is equal to ______.

पर्याय

• 7

• 8

• `15/2`

• `17/2`

Answer 1

If `lim_(n rightarrow ∞) (1^a + 2^a + ......... + n^a)/((n + 1)^(a - 1)[(na + 2) + ......(na + n)]) = 1/60` for some positive real number a, then a is equal to 7.

Explanation:

`lim_(n rightarrow ∞) (1/((a + 1)), n^(a + 1) + a_1n^a + a_2n^(a - 1) + ......)/((n + 1)^(n - 1) . n^2(a + (1 + 1/n)/2)) = 1/60`

`\implies lim_(n rightarrow ∞)  ((1/n)^a + (2/n)^a + ...... + (n/n)^a)/((n + 1)^(a - 1)[n^2a + (n(n + 1))/2]) = 1/60`

= `(lim_(n rightarrow ∞) 1/n sum_(r = 1)^n (r/n)^a)/((1 + 1/n)^(a - 1)[a + 1/2(1 + 1/n)]) = 1/60`

= `(int_0^1 x^adx)/((a + 1/2)) = 1/60`

= `(1/(a + 1))/(a + 1/2) = 1/60`

`\implies` `(1/(a + 1))/((a + 1/2)) = 1/60`

`\implies` (a + 1)(2a + 1) = 120

`\implies` 2a² + 3a – 119 = 0

`\implies` 2a² + 17a – 14a – 119 = 0

`\implies`(a – 7) (2a + 17) = 0

`\implies` a = 7, `-17/2`