Why you love series, Akul?

Algebra Level 5

Akul is immensely amused by the product of some numbers which follow a particular pattern. Mayank loves to add such "pattern-following-numbers"... Akul devises the following quantity:-

\({ k }_{ n }=\frac { (n-1).(n-3).(n-5).\quad ...\quad .\quad 2\quad or\quad 1 }{ (n).(n-2).(n-4).\quad ...\quad .\quad 2\quad or\quad 1 } \)

Mayank finds the value of \(\lim _{ n\xrightarrow { } \infty }{ (\frac { 1 }{ { e }^{ \sum _{ r=1 }^{ n }{ { k }_{ 2r } } } } } )\)

Can you guess Mayank's result?

Wanna have more fun with Mayank and Akul. This question is a part of the set Mayank and Akul

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