How do you determine whether the infinite sequence a_n=(1+1/n)^n converges or diverges?

1 Answer
Sep 22, 2014

lim_{n to infty}a_n=e (converges)

Let us look at some details.

By rewriting,

a_n=e^{ln(1+1/n)^n}=e^{nln(1+1/n)}=e^{[ln(1+1/n)]/{1/n}}

Now, let us evaluate the limit.

lim_{n to infty}a_n=e^{[ln(1+1/n)]/{1/n}} =e^{lim_{n to infty}{ln(1+1/n)]/{1/n}

by l'Hopital's Rule,

=e^{lim_{n to infty}{{-1/n^2}/{1+1/n}]/{-1/n^2}

by cancelling out -1/n^2,

=e^{lim_{n to infty}1/{1+1/n}}=e^{1/{1+0}}=e