If $\mathop \sum \limits_{r = 1}^n {a_r} = \frac{1}{6}n\left( {n + 1} \right)\left( {n + 2} \right)$ for all $n \ge 1,$ then $\mathop {\lim }\limits_{n \to \infty } \mathop \sum \limits_{r = 1}^n \frac{1}{{{a^r}}}$ is

JEE MATHEMATICS MCQ: SEQUENCE OF SERIES

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\sum \limits_{r = 1}^n {a_r} = \frac{1}{6}n\left( {n + 1} \right)\left( {n + 2} \right)

for all

n \ge 1,

then

\lim\limits_{n \to \infty } \sum \limits_{r = 1}^n \frac{1}{{{a^r}}}

View answer and explanation

Correct option: A

Option A (2) is correct because the limit of the sum converges to 2 as n approaches infinity, after simplifying the series.

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