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In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function , the Taylor polynomial is the truncation at the order k {\textstyle k} of the Taylor series of the function.
Taylor's theorem can be used to obtain a bound on the size of the remainder. In general, Taylor series need not be convergent at all. In fact, the set of functions with a convergent Taylor series is a meager set in the Fréchet space of smooth functions.
Proof. By applying an invertible affine linear change in coordinates, ... Taylor's theorem – Approximation of a function by a truncated power series; Citations
The first-derivative test depends on the "increasing–decreasing test", which is itself ultimately a consequence of the mean value theorem. It is a direct consequence of the way the derivative is defined and its connection to decrease and increase of a function locally, combined with the previous section.
Proof. If is an entire function, it can be represented by its Taylor series about 0: = =where (by Cauchy's integral formula) = ()! = + and is the circle about 0 of radius >.
Almost immediately after the superstar dropped the news of new song "The Bolter," Swifties dug up evidence it's about her ex-boyfriend.
Update, February 20, 2024: Just gonna casually circle back to this almost a year later with yet more proof that Taylor is using a cleaning supply cart to enter the arena during her Eras Tour.
In 1840, Liouville published a proof of the fact that e 2 is irrational [10] followed by a proof that e 2 is not a root of a second-degree polynomial with rational coefficients. [11] This last fact implies that e 4 is irrational. His proofs are similar to Fourier's proof of the irrationality of e.