Estimating values using the binomial theorem trivial

Question

I want to estimate a value such as $101^{1/3}$ to 4 decimal places.

I have derived the taylor expansion for the binomial $(1+x)^n$ with the remainder.

$f(x) = (1+x_0)^s + s(1+x_0)^{s-1} + ... + \frac{s(s-1)(s-2)...(s-n+1)(1+x)^{s-n}}{n!}(x-x_0)^n + R_n$

With $R_n$ = $\frac{f^{(n+1)}(c)}{n!}(x-x_0)^{n+1}$ for $x_0 < c < x$

I want to sub in $100$ into my formula but I can't since |x| < 1. I think I should do some form of a substitution for x or change my binomial.

I know I should use my remainder term and solve $R_n = 10^{-4}$ to gain the value of $n$ I should take when evaluating my sum however I do not know how to actually evaluate my sum. Also another question - what purpose does my $x_0$ actually serve in this instance, can I just set $x_0 = 0$ and use the Maclaurin expansion instead?

Answer 1

$ 101^{\frac13} = (0.808 \times 125)^{\frac13} = 5 \times (1 + (-0.192))^{\frac13}$

and now use the Binomial expansion, which is valid because $| -0.192| < 1.$