Re: binomial expansion of quantity raised to power of

• To: mathgroup at smc.vnet.net
• Subject: [mg101646] Re: [mg101598] binomial expansion of quantity raised to power of
• From: Murray Eisenberg <murray at math.umass.edu>
• Date: Sun, 12 Jul 2009 05:51:13 -0400 (EDT)
• Organization: Mathematics & Statistics, Univ. of Mass./Amherst
• References: <200907110322.XAA20134@smc.vnet.net>

```For the mathematics, look up "binomial series", e.g., at www.mathworld.com.

To get (some finite number of terms for) this in Mathematica, start with
the identity

s^2 - h^2 == s^2 (1 - (h/s)^2) // Simplify

in order to see that you can (when s > 0) obtain the new identity:

Simplify[s (1 - (h/s)^2)^(1/2) == (s^2 - h^2)^(1/2),
Assumptions -> {s > 0}]

Thus the result you cite may be obtained by:

Expand[s Normal@Series[(1 - x^2)^(1/2), {x, 0, 10}] /. x :> h/s]

Roddye Davis wrote:
> I saw in an engineering survey book a binomial expansion of (s^2 - h^2)^(1/2) = s - (h^2/2s) - (h^4/(8(s^3)))....
>
>

--
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305

```

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