The value of

\lim_{n\rightarrow \infty }\frac{1+2^{4}+3^{4}+.......+n^{4}}{n^{5}}-\lim_{n\rightarrow \infty }\frac{1+2^{3}+3^{3}+.......+n^{3}}{n^{5}}

  • Option 1)

    zero

  • Option 2)

    1/4

  • Option 3)

    1/5

  • Option 4)

    1/30

 

Answers (1)

As we learnt in 

Method of evaluating algebraic limits as x to infinity -

L=\frac{a_{\circ }}{b_{\circ }}\lim_{x\rightarrow \infty }x^{m-n}=      {0\:\:\:\:\:m<n}

                                                {\frac{a_{\circ }}{b_{\circ }}\:\:\:\:\:m=n}

                                                {\infty\:\:\:\:\:m>n}

- wherein

NOTE: for finite solution  Nr  &   Dr   degree must be same (SD : m = n )

 

 \lim_{n\rightarrow \infty } \frac{1^{4}+2^{4}+3^{4}+.....+n^{4}}{n^{5}}

\lim_{n\rightarrow \infty } \frac{1^{3}+2^{3}+3^{3}+.....+n^{3}}{n^{5}}

\lim_{n\rightarrow \infty } \frac{1}{n} \left [ \left ( \frac{1}{n} \right )^{4} + \left ( \frac{2}{n} \right )^{4} +.....\left ( \frac{n}{n} \right )^{4} \right ]

\lim_{n\rightarrow \infty } \frac{1}{n} \left [ \frac{n\left ( n+1 \right )\left ( 2n+1 \right )}{6} \right ]

\int_{0}^{1}x^{4}dx-0=\frac{x^{5}}{5}\left.\begin{matrix} & \\ & \end{matrix}\right|_{0}^{1}=\frac{1}{5}

 


Option 1)

zero

This option is incorrect. 

Option 2)

1/4

This option is incorrect. 

Option 3)

1/5

This option is correct. 

Option 4)

1/30

This option is incorrect. 

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