jee-main 2024 Q68

jee-main · India · session2_06apr_shift2 Sequences and series, recurrence and convergence Convergence proof and limit determination

$\lim _ { n \rightarrow \infty } \frac { \left( 1 ^ { 2 } - 1 \right) ( n - 1 ) + \left( 2 ^ { 2 } - 2 \right) ( n - 2 ) + \cdots + \left( ( n - 1 ) ^ { 2 } - ( n - 1 ) \right) \cdot 1 } { \left( 1 ^ { 3 } + 2 ^ { 3 } + \cdots \cdots + n ^ { 3 } \right) - \left( 1 ^ { 2 } + 2 ^ { 2 } + \cdots \cdots + n ^ { 2 } \right) }$ is equal to:
(1) $\frac { 2 } { 3 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 3 } { 4 }$
(4) $\frac { 1 } { 2 }$

$\lim _ { n \rightarrow \infty } \frac { \left( 1 ^ { 2 } - 1 \right) ( n - 1 ) + \left( 2 ^ { 2 } - 2 \right) ( n - 2 ) + \cdots + \left( ( n - 1 ) ^ { 2 } - ( n - 1 ) \right) \cdot 1 } { \left( 1 ^ { 3 } + 2 ^ { 3 } + \cdots \cdots + n ^ { 3 } \right) - \left( 1 ^ { 2 } + 2 ^ { 2 } + \cdots \cdots + n ^ { 2 } \right) }$ is equal to:\\
(1) $\frac { 2 } { 3 }$\\
(2) $\frac { 1 } { 3 }$\\
(3) $\frac { 3 } { 4 }$\\
(4) $\frac { 1 } { 2 }$

Paper Questions

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