Binomial approximation | square root expansion
ApproximationofpowersofsomebinomialsThebinomialapproximationisusefulforapproximatelycalculatingpowersofsumsof1andasmallnumberx.Itstatesthat(1+x)α≈1+αx.{displaystyle(1+x){alpha}approx1+alphax.}Itisvalidwhen|x|<1{displaystyle|x|<1}and|αx|≪1{displaystyle|alphax|ll1}wherex{displaystylex}andα{displaystylealpha}mayberealorcomplexnumbers.Thebenefitofthisapproximationisthatα{displaystylealpha}isconvertedfromanexponenttoamultiplicativefactor.Thiscangreatlysimplifymathematicalexpressions(asinthe...
Approximation of powers of some binomials
The binomial approximation is useful for approximately calculating powers of sums of 1 and a small number x. It states that
(1+x)α≈1+αx.{displaystyle (1+x){alpha }approx 1+alpha x.}It is valid when |x|<1{displaystyle |x|<1} and |αx|≪1{displaystyle |alpha x|ll 1} where x{displaystyle x} and α{displaystyle alpha } may be real or complex numbers.
The benefit of this approximation is that α{displaystyle alpha } is converted from an exponent to a multiplicative factor. This can greatly simplify mathematical expressions (as in the example below) and is a common tool in physics.[1]
The approximation can be proven several ways, and is closely related to the binomial theorem. By Bernoullis inequality, the left-hand side of the approximation is greater than or equal to the right-hand side whenever x>−1{displaystyle x>-1} and α≥1{displaystyle alpha geq 1}.
Derivations[edit] Using linea...