<s>[INST] prove increasing/decreasing sequence Are these two statements true? If so, how does one prove them? 
1) For each integer k (positive or negative), the sequence 
$a_n = (1 + k/n) ^ n$ (1) 
is increasing (at least after a certain number n).
2) For each integer k (positive or negative) the sequence 
$a_n = (1 + k/n) ^{n+1} $ (2)   
is decreasing (at least after a certain number n).
Note that I have no problem proving that the two sequences are convergent and to find their limits but I have really hard time proving formally that these are monotonic (after certain n).  
In fact for statement (1) seems one can prove it this way. 
Note that:
$(1 + \frac{k+1}{n} )^n = \frac{(1 + \frac{k}{n+1})^{n+1}(1 + \frac{1}{n})^{n}}{1+\frac{k}{n+1}}$  (3)
Now apply induction on k. Based on (3), the (k+1)-sequence ${a_n}$ must be increasing because on the right hand side we have: the increasing k-sequence ${a_{n+1}}$ multiplied by the increasing 1-sequence ${a_n}$, and then divided by a decreasing sequence. So the left hand side must be increasing too. 
Is this proof correct? Actually I think it only works for k>0 otherwise the sequence in the divisor (1 + k/(n+1)) is not decreasing but is increasing. Also, what is the proof for (2)? Should be something analogous, I guess, like the above proof for (1). 
 [/INST] Consider
\begin{align*}
\log(a_n) &= n\log\left(1 + \frac{k}{n}\right) \\
&= n\left(\frac{k}{n} - \frac{(\frac{k}{n})^2}{2} \mp \ldots  \right)\\
& = \left(k - \frac{k^2}{2n} + \frac{k^3}{3n^2} \mp \ldots \right)
\end{align*}
where the series converges absolutely for $n > k$. 
Then 
\begin{align*}
\log(a_n) - \log(a_{n-1}) &= \left(k - \frac{k^2}{2n} + \frac{k^3}{3n^2} \mp \ldots  \right) - \left(k - \frac{k^2}{2(n-1)} + \frac{k^3}{3(n-1)^2} \mp \ldots  \right) \\
&= k^2\left(\frac{1}{2n - 2} - \frac{1}{2n}\right) + O(k(k/n)^2)
\end{align*}
This is negative for $n >> k$, yielding the result. Same technique works for (2).
</s>