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Line 54: − + − File:Taylorsine.svg|The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin.+ − − File:LogTay.svg|The Taylor polynomials for {{math|ln(1 + ''x'')}} only provide accurate approximations in the range {{math|−1 < ''x'' ≤ 1}}. For {{math|''x'' > 1}}, Taylor polynomials of higher degree provide worse approximations. − − File:Logarithm GIF.gif|The Taylor approximations for {{math|ln(1 + ''x'')}} (black). For {{math|''x'' > 1}}, the approximations diverge. − −
KS update 1.4
:<math>\cosh(x) = \sum^{\infty}_{n=0} \frac{1}{(2n)!} x^{2n} \text { for all } x \!</math>
:<math>\cosh(x) = \sum^{\infty}_{n=0} \frac{1}{(2n)!} x^{2n} \text { for all } x \!</math>
:<math>e^{x} = \sum^{\infty}_{n=0} \frac{1}{n!} x^{n} = 1 + x + \frac{1}{2!} x^{2} + \frac{1}{3!} x^{3} + \cdots\text{ for all } x \!</math>
:<math>e^{x} = \sum^{\infty}_{n=0} \frac{x^{n}}{n!} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots\text{ for all } x \!</math>
:<math>\frac{1}{1-x} = \sum^{\infty}_{n=0} x^{n} = 1 + x + x^2 + x^3 + x^4 + \cdots \text{ for all } |x|<1</math>
:<math>\frac{1}{1-x} = \sum^{\infty}_{n=0} x^{n} = 1 + x + x^2 + x^3 + x^4 + \cdots \text{ for all } |x|<1</math>
File:sintay_SVG.svg|As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows {{math|sin ''x''}} and its Taylor approximations by polynomials of degree '''1''', '''3''', '''5''', '''7''', '''9''', '''11''', and '''13''' at {{math|1=''x'' = 0}}.
File:sintay_SVG.svg|As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows {{math|sin ''x''}} and its Taylor approximations by polynomials of degree '''1''', '''3''', '''5''', '''7''', '''9''', '''11''', and '''13''' at {{math|1=''x'' = 0}}.
File:Exp neg inverse square.svg|The function {{math|1=''e''(−1/''x''2)}} is not analytic at {{math|1=''x'' {{=}} 0}}: the Taylor series is identically 0, although the function is not.
File:Exp neg inverse square.svg|The function {{math|1=''e''(−1/''x''2)}} is not analytic at {{math|1=''x'' {{=}} 0}}: the Taylor series is identically {{math|0}}, although the function is not.
File:Second Order Taylor.svg|Second-order Taylor series approximation (in orange) of a function {{math|{{itco|''f''}}(''x'', ''y'') {{=}} ''ex'' ln(1 + ''y'')}} around the origin.
File:Second Order Taylor.svg|Second-order Taylor series approximation (in orange) of a function {{math|''f'' (''x'',''y'') {{=}} ''ex'' ln(1 + ''y'')}} around the origin.
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== References ==
== References ==