Find: lim n tends to infinity a^n/n! where a is constant - Maths - Limits and Derivatives

Question

Find: lim n tends to infinity a^n/n! where a is constant

Vipin Verma · Accepted Answer

limn→∞ann! ,a is a contanstAs n goes to infinity, a
stays fixed, the basic idea is that the denominator is getting
larger as compared to numerator This makes the quotient small Let x
be a integer larger then a, for instance a = 13 2 ,take x = 14, you
can take anything Suppose n>2x , thenxn<12,And we have
ann!<xnn! = (x1×x2 x2x)((x2x+1)(x2x+2)
(xn))<k(12)n-2xwhere k = (x1×x2 x2x) Notice that k does
not depend on n, but only on x Hence
limn→∞ann!≤limn→∞k(12)n-2x=limn→∞k22x(12)n=k22xlimn→∞(12)n=0Hence
limn→∞ann!=0You can check this by using a = 2, you will
get a increase upto n=2 after that the value will start decreasing
continuously211!=2, 222!=2, 233!=43, 244!=1624 it will decreases
continously untill zero

Find: lim n tends to infinity a^n/n! where a is constant.