Find the limit?
#a_1=1 & a_n=n(a_(n-1)+1)# for n=2,3,..... Define #P_n=(1+1/a_1)*(1+1/a_2)*................*(1+1/a_n)#
Then #lim_(nrarroo)P_n# is?
Then
1 Answer
Explanation:
Note that:
so:
Then, having defined:
we have:
Now if we substitute the formula for
Let now:
so that:
then:
and clearly we can establish by induction that:
and then:
Remembering that the MacLaurin formula for the exponential function is:
converging for every
and conclude that: