Use Taylor series to explain the patterns in the digits in the following expansions:
$\frac{1}{0.98}=1.02040816$
$(\frac{1}{0.99})^{2}=1.020304050607$
I was thinking that geometric series could be used but how would you use a taylor series for this problem?
Consider the function $f(x) = \frac {1}{1-x}$ evaluated at $x = 0.02$
What is the Taylor series?
$f(x) = \sum_\limits{n=0}^\infty x^n$
The Taylor series is a power series.
$f(0.02) = 0.02 + 0.02^2 + 0.02^3 + \cdots = 0.020408+\cdots$
What function might have $(\frac {1}{0.99})^2$ as its output and have a Taylor series that you think you could find?
$f(x) = \frac {d}{dx} \frac {1}{1-x} = \frac {1}{(1-x)^2} = \sum_\limits{n=1}^\infty nx^n\\ f(0.01) = 0.01020304 + \cdots$