Let `m` and `n` be two positive integers greater than 1. If
`lim_(ato0) (e^(cos(alpha^(n)))-e)/(alpha^(m))=-(e)/(2),`then the value of `(m)/(n)` is ___________.

Answer

`(2)`

Explanation

`mge2" and "nge2`
`underset(alphato0)lim(e^(cos(alpha^(n)))-e)/(alpha^(m))`
`=exxunderset(alphato0)lim(e^(cos(alpha^(n))-1)-1)/(cos(alpha^(n))-1)xx((cos(alpha^(n))-1)/((alpha^(n))^(2)))(alpha^(2n))/(alpha^(m))`
`=exxunderset(alphato0)lim((e^(cos(alpha^(n))-1)-1)/(cos(alpha^(n))-1))xxunderset(alphato0)lim((cos(alpha^(n))-1)/((alpha^(2n))))xxunderset(alphato0)limalpha^(2n-m)`
`=exx1xxunderset(alphato0)lim(-2"sin"^(2)(alpha^(n))/(2))/(alpha^(2n))xxunderset(alphato0)limalpha^(2n-m)`
`=exx1xx(-(1)/(2))xxunderset(alphato0)limalpha^(2n-m)`
Now, `underset(alphato0)limalpha^(2n-m)` must be equal to 1.
i.e., `2n-m=0`
or `(m)/(n)=2`