Binomial Expansion (P1.8 & 2.4) Flashcards
(15 cards)
describe Pascal’s triangle
adding adjacent pairs of numbers –> the numbers on the next row
the (n+1)th row of Pascal’s triangle gives the coefficients in the expansion of (a+b)^n
how do you find values of Pascal’s triangle on calculator?
nCr or (n r)
nCr = n! / r!(n-r)!
the rth entry in the nth row of Pascal’s triangle is given by (n-1)C(r-1) = (n-1 r-1)
binomial expansion with integer n values
see Wilson OneNote
why can binomial expansion be used to find simple approximations for complicated functions?
if x<1 x^n gets smaller as n increases
converges to an answer as # of terms of the expansion increases
if x is small large powers can be ignored to approximate function/estimate a value
when is binomial expansion valid/when estimation works?
|x|<1
-1 < x < 1
when are approximations based on binomial expansion more accurate?
when more terms of the expansion are used
when the values of x substituted into the expansion are closer to 0
when is the expansion of (1+bx)^n valid?
when |bx| < 1
|x|< 1/|b|
expanding (1+x)^n when n is not a +ve integer
use formula book
when n is not a natural number (+ve integer)…
none of the coefficients = 0
so this version gives an infinite # of terms
what is the method to expand (a+bx)^n?
a^n (1 + b/a x)^n
when is the expansion of (a+bx)^n valid?
when |b/a x| < 1
|x|<|a/b|
what is the method for binomial expansion with partial fractions?
- split into partial fractions
- work out each expansion separately
- add or subtract the 2 expansions
what is the method for using a substitution of x to find an approximation for a given value?
- substitute in value for x into the expanded & non-expanded form
- make the non-expanded form into the given value, doing the same to the expanded side
when doing binomial expansion with integer value for n, 1st expansion term starts with nC0 not nC1
the nth entry in the rth row of Pascal’s triangle is given by
(n-1)C(r-1)
