Taylor series applications Watch

Big-Daddy
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Can we write any trig function (e.g. sin, cos, tan, cot, sec, sinh, cosh, tanh, arcsin, arctan etc.) as well as any a^(f(x)) as a Taylor series? i.e.

\displaystyle \sum_{i=0}^n a_i x^i

Where n could be infinity.

If so, how do we find the Taylor series rewriting of a^(f(x))?
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Jkn
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(Original post by Big-Daddy)
Can we write any trig function (e.g. sin, cos, tan, cot, sec, sinh, cosh, tanh, arcsin, arctan etc.) as well as any a^(f(x)) as a Taylor series? i.e.

\displaystyle \sum_{i=0}^n a_i x^i

Where n could be infinity.

If so, how do we find the Taylor series rewriting of a^(f(x))?
Firstly remember that sinh, cosh, etc.. are not trigonometric functions, they are hyperbolic functions. The are similarly named as they share certain properties as well as particularly elegant relations when it comes to considering complex values.

Secondly, all Taylor series can be expressed as an infinite number of terms (i.e. n \to \infty) though there is the special case of polynomial functions whereby a polynomial of degree d has a_i=0 for all i>d (i.e. the Taylor series of a polynomial is intact itself!)

Thirdly, what you have presented here is a special case of the Taylor's series (the Macluarin's Series) which is a de-generalisation of \displaystyle \sum_{i=0}^{\infty} a_i (x-b)^i in that it considers only the case where b=0.

Fourthly, any infinitely differentiable function has a Taylor's series (which should answer your first question in that it is valid so long as a^{f(x)} is infinitely differentiable).

Fifthly, finding that Taylor series follows directly from the definition.

I believe the reason no-one replied to you (up until now) is that all your questions can be answered by actually reading the one-line definition for a Taylor's series. I suggest you give these things a go first and then, if you get stuck, use TSR to find someone who can explain things for you. Also, if you are struggling to find resources, it's always worth trying wikipedia and the like (despite their negative reputation they are actually good!)
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