# Thread: Converging Sequence!! AKA Infinite Series!!

1. ## Converging Sequence!! AKA Infinite Series!!

Question:

Show that the sequence is convergent by computing its limit when n -> ∞

a(n) = (e^-n)/[1-n^(1/n)]

Related Equations:

lim [n -> ∞] a(n) = L

My Attempt:

I tried using l'hopitals rule however it does not work.
Any suggestions???

2. Originally Posted by olyviab1
Question:

Show that the sequence is convergent by computing its limit when n -> ∞

a(n) = (e^-n)/[1-n^(1/n)]

Related Equations:

lim [n -> ∞] a(n) = L

My Attempt:

I tried using l'hopitals rule however it does not work.
Any suggestions???

Hint: since clearly $\displaystyle n>\left(1+\frac{1}{n}\right)^n\,\,\,\forall\,3<n\i n\mathbb{N}$ ,we get $\displaystyle \sqrt[n]{n}>1+\frac{1}{ n}$ , and thus:

$\displaystyle 0\leq \frac{e^{-n}}{n^{1\slash n}-1}\leq \frac{e^{-n}}{1\slash n}$ $\displaystyle =\frac{n}{e^n}$ , and this last limit is zero by L'H. rule ... and yes, I know the denominator is twisted, but it never minds.

Tonio