Question

`lim_(n→∞)((1^2 + 2^2 + ...... + n^2)(1^4 + 2^4 + ...... + n^4))/((1^7 + 2^7 + ......  n^7)) = (k + 1)/15`, then k is equal to ______.

Options

• 6.00

• 7.00

• 8.00

• 9.00

Answer 1

`lim_(n→∞)((1^2 + 2^2 + ...... + n^2)(1^4 + 2^4 + ...... + n^4))/((1^7 + 2^7 + ......  n^7)) = (k + 1)/15`, then k is equal to 7.00.

Explanation:

`lim_(n→∞)(((1^2 + 2^2 + ...... +  n^2)/n^3)((1^4 + 2^4 + ...... + n^4)/n^5))/((1^7 + 2^7 + .... + n^7)/n^8)`

= `((int_0^1x^2dx).int_0^1x^4dx)/(int_0^1x^7dx)`

= `8/15`

= `(k + 1)/15`

⇒ k = 7