factorials, square roots, and infinite series

a few math questions
in the binomial expansion theroem equation, basically:

Code:

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(a+b)!
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a! * b!

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where a+b is the exponent to multiply out a (x+y) binormial, and b+1 is the b'th element in that line in pascal's triangle

how is that equation derived?
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is there a way to get a square root of a non-square number without a bunch of trial and error?
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in an infinite geometric series, the equation for the sum is:

Code:

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a
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1-r

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where r is between 1 and -1, and a is the first element. how is this equation derived? i can only get this far:

Code:

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sum = a + a*r + a*r^2 + a*r^3 ...
    = a(1 + r + r^2 + r^3 ...)

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A) Is this calc, cause this **** is crazy.:eek:

B) How old are you?

a) pre-calc
b) 17

on the first question, i just want to know how ppl derived the binomial expansion theorem.

oh, and fyi, this isn't any kind of homework... i'm just curious

Non perfect squares have irrational roots so it depends on how accurate you want to be and if you're after quick easy calculations for a few decimal places on paper or an algorithm for as many as you can be bothered or you want a PC to use, remembering of course the limit of the finite number set on a PC.

A reasonable approximation for a few decimal places or so for numbers greater > 1 follows presented as an example for simplicity.

What is the square root of 27?

Now 25^1/2 = 5, and 36^1/2 = 6

let 5+x = 27^1/2 where x is less than 1.
27 = (5+x)^1/2
27 = 25 +10x +x^2
Since x < 1, x^2 is "small" so ignore it.
27 ~ 25 + 10x
x ~ 0.2

Giving 27^1/2 = 5.2
5.2^2 = 27.04

Algorithm-wise, try looking at math libraries. I only one I looked at was for the sine function and I can't remember if it were just trig functions or a more general maths algorithms, or the URL and have no incentive to go look for it either.

lol. im 15 in ninth grade and this stuff is EASY! i just finished algebra II

Quote:

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Originally posted by Leeman_s
lol. im 15 in ninth grade and this stuff is EASY! i just finished algebra II

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Me too :D only I'm in calculus....

let me restate my square root question:
pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - .......
pi is an irrational number
is there a similar arrangement for irrational square roots?

Damn, I'm in Pre-Calc and we are still in trig, using trig identities to solve trig equ's. But, we only have the courses for half a year cause my school is dumb.