Homework: Examples for Special Sequences Sorry for asking this question. But wrecked my brain and having trouble coming up with anything. 

Find real sequences $(a_n), (b_n)$ and  $(c_n)$ such that $\lim a_n
 = \lim b_n  = \lim c_n  = 1$ but $\lim (a_n)^n = 0$, $\lim(b_n)^n = 2014$ and $\lim(c_n)^n = \infty$

The only one that is obvious to me is $b_n = \sqrt[n]{2010}$. Can't come up with examples for the rest. Any help would be appreciated. 
 A: Focus on the second conditions first. What kinds of sequences can you find such that $(x_n^n)$ tends to $0$, to $2014$ (ie. to a large positive real number), or to $\infty$?
For the second one, $x_n^n\to2014$, the obvious choice is $\sqrt[n]{2014}$, which converges to $1$. So can we find sequences for the other two conditions that also converge to $1$?
Well, for $x_n^n\to0$, you should be able to see immediately that we'll need $(x_n)$ to stay in the range $(-1, 1)$. We have $a^n\to 0$ for any $a$ in that range, but the convergence is slower and slower as $a$ approaches $1$ (look at some graphs of $x^n$ for large $n$. Note the sharp increase as you get near $1$). Hmm, but we specifically want $(x_n)$ to converge to $1$. Can we get it to converge slowly enough that $x_n^n$ still goes to $0$?
For $x_n\to\infty$, we have a similar problem. We have $a^n\to\infty$ for any $a>1$, but more and more slowly as $a$ approaches $1$. So you'll need to find an $(x_n)$ that converges slowly enough to $1$ that raising it to the $n$-th power is still able to "beat" that effect. Note that if you can work out an $(x_n)$ that converges to $0$ for the first part, you can easily use it to obtain another $(x_n)$ that converges to $\infty$.
A: As rogerl suggested, $(a_n)$, $(b_n)$ and $(c_n)$ should have the following shape $1+(d_n)$ where $\lim d_n  = 0 $ 
Hint : 
$$ 
(1+d_n)^n = \exp(n \cdot \ln(1+d_n)) = \exp(n \cdot (d_n-d_n^2/2+o(d_n^2)))
$$
