Approximating radicals using Binomial theorem

Question

Use a suitable binomial expansion to find square root of $1.01$ and correct it to five decimal places. I use the formula

$$(1+ax)= 1+ax + \frac{a(a-1)}{2!} + \cdots$$

but do not know where to stop.

Answer 1

At each step, the coefficient multiplies by a certain amount. Past a certain point you can tell that the coefficient will always be less than some $c$ in magnitude, so the remainder would have magnitude less than $c ( x^k + x^{k+1} + \cdots ) = c \dfrac{x^k}{1-x}$. Using that you can determine when you are close enough.