Recap on Power Series, Taylor Series, and Taylor Polynomials

mes (67)in DTube • 5 days ago

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

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