Here is a simple way to understand the binomial approximation. Consider (1 + x )
ID: 1511313 • Letter: H
Question
Here is a simple way to understand the binomial approximation. Consider (1 + x)2 = (1 + x)(1 + x). If you multiply this out, you get (1 + x)2 = 1 + 2x + x2. Now, if lxl << 1, then x2 << x, so we have (1+x)2 1 + 2x, just as the bionomial approximation states. Apply the same kind of reasoning to (1 + x)3 and (1 + x)4, and show that the bionomial approximation works in these cases, too, when x is small enough that we can ignore x2 (and higher powers of x) compared to x. (While this is not a proof, it may help you understand the basic issues involved.)
Explanation / Answer
(1+x)3 = 1+3x + 3x2 + x3
now x2 and x3 are << x
So, ignoring them, (1+x)3 = 1 + 3x
(1+x)4 = (1+x)2*(1+x)2 = (1+2x)2 = (Put y = 2x)
(1+y)2 = 1 + 2y = 1 + 4x