Taylor Series and Maclaurin Series - Calculus 2

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3:00For ln x around $x=1$, derivative values give ($x-1$)−($x-1$)²/2+($x-1$)³/3−⋯. Equality with the function holds for |$x-1$|<1; endpoints require separate checking.

13:30At zero, the even-power coefficients of sine vanish and the odd-power terms alternate: $x-x$³/3!+x⁵/5!−⋯. This series converges to sine for every real x.

24:00Termwise differentiation gives the cosine series; replacing x by x² then gives $1-x$⁴/2!+x⁸/4!−⋯ for cos(x²). These operations are justified within the power series’ convergence range.

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