Recap on Power Series, Taylor Series, and Taylor Polynomials

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In this video I go over a quick recap on power series, which is a specific type of infinite series whose terms are powers of x multiplied by coefficients. A power series converges for only 3 possible cases: at x = a, at all values of x, or at an interval of convergence R while the endpoints are indeterminant. A Taylor series is a specific type of power series whose coefficients are determined from the coefficients of a function. Taylor polynomials refer to a specific number of terms of a Taylor series. If a function is equal to its corresponding representation as a Taylor series then the remainder approaches zero. Lastly, a Maclaurin series is a specific type of Taylor series that is centered about zero.

Timestamps:

• Power Series: 0:00
  • Convergence of a power series at x = a, all x, or a radius of convergence R: 0:56
  • Series may or may not converge at end points of R: 1:42
• Taylor series is a power series with coefficients obtained from the function's derivatives: 2:37
  • Maclaurin series is a Taylor series centered at a = 0: 3:23
  • A function is equal to its Taylor series if remainder approaches zero: 3:55
• Taylor's Inequality is used to evaluate the remainder: 4:24
• Table 1 of common functions and their Maclaurin series: 5:13

Full video, notes, and playlists:

• Full video and playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0F76sIU8xm09oqBTq1mlry3
• HIVE notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-applications-of-taylor-polynomials
• Infinite Sequences and Series: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz .

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