Can binomial have negative power?

The binomial theorem for positive integer exponents n can be generalized to negative integer exponents. This gives rise to several familiar Maclaurin series with numerous applications in calculus and other areas of mathematics.

How are binomial expansions related to Pascal’s triangle quizlet?

The binomial expansion uses Pascal’s triangle for the corresponding coefficient for each row that corresponds to the number that the expansion term is raised to. You just studied 7 terms!

How are the exponents of a binomial expansion related?

Exponents of each term in the expansion if added gives the sum equal to the power on the binomial. The powers of the first term in the binomial decreases by 1 with each successive term in the expansion and the powers on the second term increases by 1.

Is there a way to recover the negative binomial theorem?

Negative Binomial Theorem. While positive powers of 1+x can be expanded into polynomials, e.g. (1+x)3 = 1+3x+3×2 +x3, f (x) cannot be, so there cannot be a finite sum of monomial terms that equals f (x) . But there is a way to recover the same type of expansion if infinite sums are allowed. As a first approximation,…

How is the binomial theorem generalized to negative integer exponents?

The binomial theorem for positive integer exponents \\( n \\) can be generalized to negative integer exponents. This gives rise to several familiar Maclaurin series with numerous applications in calculus and other areas of mathematics. \\( f(x) = (1+x)^{-3} \\) is not a polynomial.

Is there a way to recover the same type of expansion?

While positive powers of f (x) f (x). But there is a way to recover the same type of expansion if infinite sums are allowed. y = 1 -3x y = 1 −3x. So for small 1 ( 1 + x) 3 ≈ 1 − 3 x. ≈ 1− 3x. This approximation is already quite useful, but it is possible to approximate the function more carefully using series.