I have to use a binomial expansion to evaluate $1/\sqrt{4.2}$ to $5$ decimal places. The answer from a calculator is $0.48795$ but I get $0.48202$, so I'm doing something wrong. I've also checked my calculations on a spreadsheet so the problem is with my technique, not arithmetic. Here's what I'm doing:
Using
$$
(x + y)^{-n} = x^{-n}\sum_{k=0}^\infty {^{-n}}C_k \left( \frac y x \right)^k
$$
I need
$$
(4 + 0.2)^{-1/2} = 4 ^{-1/2} \sum_{k=0}^\infty {^{-n}}C_k \left( \frac {0.2} {4} \right)^k
$$
$$
= \frac1 2 \sum_{k=0}^\infty {^{-n}}C_k \left( \frac {1} {2} \right)^k \left( \frac {1} {10} \right)^k
$$
Using the recurrence relation
$$
^nC_{k+1} = \frac{n-k}{k+1} {^n}C_k
$$
I caluculate
$^{-1/2}C_0=1$;
$^{-1/2}C_1=-3/4$;
$^{-1/2}C_2=5/8$;
$^{-1/2}C_3=-35/64$
So the evaluation should be:
$$
\frac 1 2\left( 1 + \frac {-3}{4}.\frac1 2.\frac 1 {10} + \frac {5}{8}.\frac1 4.\frac 1 {100} + \frac {-35}{64}.\frac1 8.\frac 1 {1000} \right)
$$
but this is incorrect, as described above. It is enough terms because the last one is $-0.0000342$.
What am I doing wrong?
Best Answer
${}^{-1/2}C_1=\frac{-\frac12-0}{0+1}\cdot {}^{-1/2}C_0$ etc.
Also, the last term should be $<10^{-5}$ to be sure about the 5 places.