Some kind of mathematical formula that I obviously dont know about?

[M. next]

Hello there.
Can you please explain the following:
Let us suppose that δ is very much less than r, and expand |r+δ|^-1 in powers of δ/r. Keeping the lower order terms gives:
|r+δ|^-1 ≈ 1/r.(1-r.δ/r^2) + 1/2r [3(δ.r/r)^2-(δ/r)^2]

How did they reach here? is this kind of a mathematical formula that i don't know about?

Thanks in advance

 

[rlewit]

I don't know that your aproximation is correct? If I set δ = 0 then the equation becomes

1/|r| ≈ 1/r

That doesn't look right?

 

[mathman]

|r+δ| = |r||1+δ/r|. Since δ/r << 1, you may use |r|(1+δ/r).

Take the reciprocal and expand (1+δ/r)^-1 in a power series.

 

[M. next]

What do you exactly mean by a power series, mathman?

 

[NascentOxygen]

What do you exactly mean by a power series, mathman?

See the binomial series, and use just the first few terms, since they diminish rapidly for d/r very much smaller than 1: http://mathworld.wolfram.com/BinomialSeries.html

 

[NascentOxygen]

I would check the approximation formula that you quote, too. That is, evaluate it for a small delta/r, say, .0015, and verify that it gives a good approximation. (Otherwise you might be futilely trying to arrive at a formula that you have transcribed wrongly. I haven't checked it.)

 

[M. next]

Thank you, NascentOxygen. I get it now. :)

 

[mathman]

(1+x)^-1 = 1 - x + x² - x³ + x⁴ - ...

Formula holds for |x| < 1.

 

[M. next]

Thank you.