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This video explains how to use Taylor series and power series to approximate functions. The author shows how to write code in Python and Matlab to compute the expansion, and how the approximation gets better as the order of the series is increased.

**00:00:00**Taylor series is a series approximation for a function. It is important in calculus, and can be used to solve differential equations and other problems. In this video, I explain Taylor series in high-level terms and show how to use it in Python and Matlab.**00:05:00**In this video, Taylor series and power series are explained in detail. The Taylor series expansion is valid for any Delta X, but will converge until a discontinuity is reached. The Maclaurin series is a special case of the Taylor series, where the base point is equal to zero.**00:10:00**Taylor series and power series are both types of expansions, and both are easy to understand with pictures. The Taylor series for sine of x is 0 cosine of zero plus the derivative of cosine at zeros. The power series for cosine of x is cosine of zero plus the derivative of cosine at zeros, plus the second derivative of cosine at zero, plus the third derivative of cosine.**00:15:00**The Taylor series is a series of mathematical functions that are easy to compute when the functions have a low degree of complexity. The linear, cubic, fifth, seventh, and ninth order approximations of the sine function are plotted in Matlab and Python.**00:20:00**The author demonstrates how Taylor series and power series can be used to approximate functions. He writes code in Python and Matlab, and shows how the approximation gets better and better as the order of the series is increased.**00:25:00**In this video, the author shows how to approximate a function using a Taylor series. First, they define the coefficients of the polynomial and then plot the linear, cubic, and quintic functions on top of each other to see if they get better and better. The first order Taylor expansion is a smooth function and all of its derivatives exist. As the author adds more terms, the approximation gets better and better.

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