-
The Taylor Series is defined as
Where
is the -th derivative of and . -
When
in the above, we call it a Maclaurin Series. -
Partial sums are called Taylor Polynomials. This gives a
-th order approximation. Typically, we then have, assuming Taylor polynomial To be more precise
Where
is the error term obtained from the rest of the sequence. We denote the order with Big Oh Notattion
Where
is the constant, and is the order of the approximation. -
Every Taylor Series has a domain of convergence within which it gives good approximations.
Common Maclaurin Series
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The Exponential Function
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The Natural Logarithm (first equation) and the Mercator Series (second equation)
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The geometric series
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The binomial series
Where
If we want
we can use the above as -
The square root and its inverse
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The Sine and Cosine functions
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The Inverse Sine and Cosine functions for
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The inverse tangent function for
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The hyperbolic sine and cosine
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The inverse hyperbolic functions for