If lim n → ∞ ∑ r = 1 2 n 3 r 2 n 3 e r 3 n 3 = e a - e b , then a + b is equal to

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Question: Answered & Verified by Expert

If

\lim_{n \to \infty} \sum_{r = 1}^{2 n} \frac{3 r^{2}}{n^{3}} e^{\frac{r^{3}}{n^{3}}}

= e^{a} - e^{b}

, then

a + b

is equal to

MathematicsDefinite IntegrationJEE Main

Solution:

2925 Upvotes Verified Answer

The correct answer is: 8

The given limit is

$\lim_{n \to \infty} \sum_{r = 1}^{2 n} \frac{3 r^{2}}{n^{3}} e^{\frac{r^{3}}{n^{3}}} = \int_{0}^{2} 3 x^{2} e^{x^{3}} d x$
$= {[e^{x^{3}}]}_{0}^{2}$
$= e^{8} - 1 = e^{8} - e^{0}$

Hence, $a + b = 8$

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