How do I see that $1-(1-\frac 1 M)^Q \approx \frac Q M$ (provided that $Q$ is small compared to $M$), where both $Q$ and $M$ are integers?

Question

How do I see that $$1-(1-\frac 1 M)^Q \approx \frac Q M$$ (provided that $Q$ is small compared to $M$), where both $Q$ and $M$ are integers ?

The approximation is stated in a book without proof.

I've tried looking into some analysis and using the binomial expansion theorem, but haven't come to a conclusion.

Also, how "small" should $Q$ be compared to $M$ before the approximation is good ?

Answer 1

Another way to look at it compared with clement nice comment/answer is $$ (x+y)^n = \sum_{k=0}^{n}\left(\matrix{n\\k}\right)x^{k}y^{n-k} $$ Putting in $x= -1/M$ and $y=1$ for convience. We find $$ (-\frac{1}{M} +1)^Q= \sum_{k=0}^{Q}\left(\matrix{Q\\k}\right)\left(-\frac{1}{M}\right)^{k} $$ The first few terms are $$ \left(\matrix{Q\\0}\right)\left(-\frac{1}{M}\right)^{0} + \left(\matrix{Q\\1}\right)\left(-\frac{1}{M}\right)^{1} + ... $$ Which yields your result. The size of $x$ in this case determines you can truncate the binomal expansion to first order terms.

Answer 2

Consider the geometric progression: $$ \overbrace{\left(1{}+{}\left(1-\frac{1}{M}\right){}+{}\ldots{}+{}\left(1{}-{}\frac{1}{M}\right)^{Q-1}\right)}^{Q\mbox{ terms}}{}={}\frac{\left(1{}-{}\left(1{}-{}\frac{1}{M}\right)^Q\right)}{\left(1{}-{}\left(1{}-{}\frac{1}{M}\right)\right)}{}={}M\left(1{}-{}\left(1{}-{}\frac{1}{M}\right)^Q\right) $$

For $M$ sufficiently large, each of the $Q$ terms on the l.h.s is approximately $1$. Therefore, their sum in this case is approximately $Q$, giving $$ \frac{Q}{M}\approx\left(1{}-{}\left(1{}-{}\frac{1}{M}\right)^Q\right)\,. $$ For $Q$ and $M$ not integers, the argument required is more sophisticated.