$$\mathop {\lim }\limits_{n \to \infty } {{1 + {2^4} + {3^4} + .... + {n^4}} \over {{n^5}}}$$ - $$\mathop {\lim }\limits_{n \to \infty } {{1 + {2^3} + {3^3} + .... + {n^3}} \over {{n^5}}}$$
JEE · Math · previous-year question
- A.$${1 \over 5}$$correct
- B.$${1 \over 30}$$
- C.zero
- D.$${1 \over 4}$$
Answer
A. $${1 \over 5}$$
Explanation
The given expression can be written as $$\mathop {\lim }\limits_{n \to \infty } {1 \over n}{\sum\limits_{r = 1}^n {\left( {{r \over n}} \right)} ^4} - \mathop {\lim }\limits_{n \to \infty } {1 \over n}.\mathop {\lim }\limits_{n \to \infty } {1 \over n}{\left( {{r \over n}} \right)^3}$$ $$ = \int\limits_0^1 {{x^4}} \,\,dx - \mathop {\lim }\limits_{n \to \infty } {1 \over n} \times \int\limits_0^1 {{x^3}} \,\,dx$$ $$ = \left[ {{{{x^5}} \over 5}} \right]_0^1 - 0 = {1 \over 5}$$
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