Question

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

Answer 1

`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`