Binomial Series

Calculus 2 · Axiom Academy

Extending the binomial theorem past whole-number exponents to get infinite-series expansions for roots and fractional powers. 1. The Classical Binomial Theorem First recall what we already have. For any positive integer n and any real x , the expansion is a finite sum of exactly n+1 terms: A finite sum — it stops at the x^n term For example (1+x)^3 = 1 + 3x + 3x^2 + x^3 . The animation lays the four terms out and shows the wall where the expansion terminates . 2. The Generalized Binomial Coefficient To allow any real exponent k , we need a coefficient that makes sense even when k isn't a whole number. The fix: replace n!/(k!(n-k)!) with the falling factorial — a product of n factors that each step down by one, divided by n! . Start at k and multiply n factors, each one less than the last: The animation builds this descending product for . Now we can state the full theorem. Use the generalized coefficient and let the sum run to infinity: Critical condition: the series converges only when . Below, the dashed curve is the true (1+x)^ 1/2 . Watch the partial sums lock onto it inside the band — and peel away outside it. Finite sum. Works for every x . 4. Application: the Square-Root Series The most useful case is , giving a series for . Reading off the falling-factorial coefficients: Each extra term shrinks the gap to the true value. The animation fixes a test point x = 0.6 and shows the error bar collapsing as we go . Take x = 0.1 and keep the first three terms:

This is the written version of the interactive lesson above. See the full Calculus 2 course.