Beyond the Binomial Theorem: The Binomial Series

►Get my favorite, free calculator app for your phone or tablet: MAPLE CALCULATOR: https://www.maplesoft.com/products/maplecalculator/download.aspx?p=TC-9857 ►Check out MAPLE LEARN for your browser to do compute and graph math: https://www.maplesoft.com/products/learn/?p=TC-9857 ►Check out MapleSoft's YouTube channel: https://www.youtube.com/channel/UCq2MmZQ8-kqEVAnmL2GSMsQ. Thank you to MapleSoft for sponsoring today's video. In this video we will be building up to the Binomial Series. We start with Pascal's Triangle, whose coefficients are found in the expansion for powers of binomials. We then take a combinatorial approach to come up with the Binomial Theorem which applies to positive integer powers of binomials. But what if the exponent is some other real number? Well we can use the power of Taylor Series to come up with the Binomial Series, and along the way we define a natural extension of the "n choose k" notation that work for arbitrary real numbers too. 0:00 Pascal's Triangle 0:51 Expanding Binomials with Maple Calculator 2:00 Connecting Coefficients to Pascal's Triangle 4:03 Combinatorical Approach to Binormial Coefficients 5:12 Binomial Theorem 6:23 Extending "n choose k" formula 8:22 Taylor Series Review 10:00 The Binomial Series 11:22 Binomial Series in Maple Learn Check out my MATH MERCH line in collaboration with Beautiful Equations ►https://www.beautifulequation.com/pages/trefor COURSE PLAYLISTS: ►DISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS ►LINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6 ►CALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m ► CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n ►MULTIVARIABLE CALCULUS (Calc III): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd ►VECTOR CALCULUS (Calc IV) https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa ►DIFFERENTIAL EQUATIONS: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxde-SlgmWlCmNHroIWtujBw ►LAPLACE TRANSFORM: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxcJXnLr08cyNaup4RDsbAl1 ►GAME THEORY: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxdzD8KpTHz6_gsw9pPxRFlX OTHER PLAYLISTS: ► Learning Math Series https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw ►Cool Math Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho BECOME A MEMBER: ►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join SOCIALS: ►Twitter (math based): http://twitter.com/treforbazett ►Instagram (photography based): http://instagram.com/treforphotography