Expert: Abe Mantell - 5/25/2009

Question
Use the binomial series to approximate √6 to within .005

Answer
Hello Laurel,

Sorry for my delay with this...had e-mail problems and did not see this one!

The binomial theorem, for n a natural number,

(a + b)^n = a^n + n a^(n-1) b + n (n-1) a^(n-2) b^2 / 2 + ... + (n! / ((n - i)! i!) a^(n - i) b^i + ... + b^n.

may be proven by Mathematical Induction.

If we set a equal to one and replace b by x, we obtain the power series

(1 + x) ^ n = 1 + n x + n (n - 1) x^2 / 2 + n (n - 1) (n - 2) x^3 / 6 + ... (n! / ((n - i)! i!) x^i + ... + x^n.

By additional Mathematical Induction, it may be shown that this binomial expansion holds for any rational n; however, the series becomes infinite, with a radius of convergence abs(x) < 1. Because of the uniqueness of derivatives, this series is the Taylor series.

For example, take n = 1 / 2 to obtain the Taylor series for a square root

sqrt(1 + x) = (1 + x) ^ (1 / 2) = 1 + (1 / 2) x - (1 / 8) x ^ 2 + (1 / 16) x ^ 3 + ... + ((- 1) ^ i (i - 3 / 2)! / ((- 3 / 2)! i!)) x ^ i + ...

Now, for your problem of sqrt(6), we need to get closer to 6...so use
the expansion for (4 +x), since 4 is the closest perfect square to 6.

I'll leave to you to work ot the details...then let x=2 to get an
approx. to sqrt(6).

Abe