Question

`lim_(n rightarrow ∞) (1^4 + 2^4 + 3^4 + ...n^4)/n^5 - lim_(n rightarrow ∞) (1^3 + 2^3 + 3^3 + ...n^3)/n^5` is ______.

Options

• `1/5`

• `1/30`

• Zero

• `1/4`

Answer 1

`lim_(n rightarrow ∞) (1^4 + 2^4 + 3^4 + ...n^4)/n^5 - lim_(n rightarrow ∞) (1 + 2^3 + 3^3 + ...n^3)/n^5` is `underlinebb(1/5)`.

Explanation:

`lim_(n rightarrow ∞) (1^4 + 2^4 + 3^4 + ......n^4)/n^5 - lim_(n rightarrow ∞) (1^3 + 2^3 + 3^3 + ......n^3)/n^5`

= `lim_(n rightarrow ∞) 1/n sum_(r = 1)^n (r/n)^4 - lim_(n rightarrow ∞) 1/n . lim_(n rightarrow ∞) 1/n (r/n)^3`

= `int_0^1 x^4 dx - lim_(n rightarrow ∞) 1/n xx int_0^1 x^3 dx`

= `[x^5/5]_0^1 - 0`

= `1/5`