When is it not safe to apply the approximation (1+a)^N = 1+Na (for a<<1)?

Question

For example, consider the following equation where $|k|<<1$, $N$ is a positive integer ranging from 1 to 100,000) and $k,N$ are both real. $$ \left(\frac{1+k}{k} \right)\left[(1+k)^N-1 \right] $$ Route (1) using the (?Bernoulli?) approximation immediately we get $$ \left(\frac{1+k}{k} \right)\left[(1+Nk)-1 \right] \rightarrow \left(\frac{1+k}{k} \right)Nk \rightarrow (1+k)N \rightarrow N+kN. $$ Route (2) rearranging before using the approximation we get $$ \left(\frac{1}{k} \right)\left[(1+k)^{N+1}-(1+k) \right] \rightarrow \left(\frac{1}{k} \right)\left[(1+Nk+k)-(1+k) \right] \rightarrow \frac{Nk}{k} \rightarrow N. $$ Clearly use of the approximation in this particular case gives ambiguous results. But are there general guidelines on when it is or isn't safe to use this approximation - either in examples similar to the one above or, more generally, to a wider range of problems?

Answer 1

(I'm assuming $N\ge1$ and that $k$ is positive.) I wouldn't try to use this approximation when $Nk>1$. When $Nk \le 1$, it's true that $(1+k)^N = 1 + Nk + O(N^2k^2)$ - in other words, there exists a constant $C$ such that $|(1+k)^N - (1+Nk)| \le CN^2k^2$. So the approximation is best when $k \ll 1/N$, not just $k \ll 1$.

This also helps to explain why the two answers are compatible: if you carry the error terms when you approximate, you'll see that the two approximations are $N+kN + O(N^2k)$ and $N+O(N^2k)$, respectively. In other words, the potential error from the approximation is greater than $kN$, so that term isn't significant.