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FP2: Taylor's Series

which of the equations for Taylors series fro I use and what's my value for a and why?
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Original post by Maths&physics
which of the equations for Taylors series fro I use and what's my value for a and why?


Isn't there only one for Taylor series?

Expansion about the point x=ax=a is:

f(x)=f(a)+f′(a)(x−a)+12f′′(a)(x−a)2+16f′′′(a)(x−a)3+…f(x) =f(a) + f'(a)(x-a) + \dfrac{1}{2} f''(a)(x-a)^2 + \dfrac{1}{6}f'''(a)(x-a)^3 + \ldots
Original post by Maths&physics
which of the equations for Taylors series fro I use and what's my value for a and why?


There is only one Taylor's Series for the function sec^2x.

Which is ( f(x) evaluated at a) ∑n=0∞f(n)(a)n!(x−a)n \displaystyle \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n and we call this Maclaurins series when evaluated at a=0.

Notice that in Taylor's series we use (x−a) (x-a) so what is a equal to in (x−π4) (x-\frac{\pi}{4}) ?
(edited 5 years ago)
Original post by RDKGames
Isn't there only one for Taylor series?

Expansion about the point x=ax=a is:

f(x)=f(a)+f′(a)(x−a)+12f′′(a)(x−a)2+16f′′′(a)(x−a)3+…f(x) =f(a) + f'(a)(x-a) + \dfrac{1}{2} f''(a)(x-a)^2 + \dfrac{1}{6}f'''(a)(x-a)^3 + \ldots


there are different types on here: https://www.examsolutions.net/tutorials/taylors-series/?level=A-Level&board=Edexcel&module=FP2&topic=1926
Original post by NotNotBatman
There is only one Taylor's Series for the function sec^2x.

Which is ( f(x) evaluated at a) ∑n=0∞f(n)(a)n!(x−a)n \displaystyle \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n and we call this Maclaurins series when evaluated at a=0.

Notice that in Taylor's series we use (x−a) (x-a) so what is a equal to in (x−π4) (x-\frac{\pi}{4}) ?


thanks, how many different types are there?
Original post by Maths&physics
thanks, how many different types are there?


The Taylor's series for a one variable function is given by the formula aforementioned, there isn't another type.
Original post by Maths&physics


No, there is only one Taylor's series.

If a=0a=0 then it's a special case of MacLaurin's series.
Original post by Maths&physics


The point of that second example is to show that for some really small value epsilon ∣a∣−ϵ=0 |a| - \epsilon = 0 you can use the taylor's series to approximate a function very close to 0, if the Maclaurins series fails (as some g(n)(0)n! \frac{g^{(n)}(0)}{n!} may not be defined). When asked to use the Taylor's series you use the one mentioned throughout this thread.

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