Hello I have some problems concerning Taylor series.

Given the function $$f(x)=e^{\sin{x}} $$

I concluded that the Taylor series expansion would be

$$f(x) = \sum^\infty_{n=0}\frac{1}{n!}f^{(n)}(x)(x-x_0)^n$$ But Wolfram wrote a much simpler form


My question is: how come? Secondly, could somebody explain to me how to get the second sum's radius of convergence?

Thank you very much for your time!

  • $\begingroup$ You might need to revise your conception of what a Taylor series is saying: the factor $f^n(x)$ should probably read $f^{(n)}(x_0)$ (thus, two mistakes). $\endgroup$ – Did Mar 8 '14 at 15:42
  • $\begingroup$ @Did of course you're right - I didnt write the parentheses and followed my previous statement mindlessly - edited $\endgroup$ – Simon Mar 8 '14 at 15:45
  • $\begingroup$ Worlfram's series, while equal to your function $f$ (by series composition), is not a power series, so it isn't the Taylor series of anything, doesn't have a radius, etc. $\endgroup$ – Andrew D. Hwang Mar 8 '14 at 15:49

Take in to major account Did's comment. If you properly apply the definition and build your series at $x=0$, you will obtain $$1+x+\frac{x^2}{2}-\frac{x^4}{8}-\frac{x^5}{15}-\frac{x^6}{240}+\frac{x^7}{90}+\frac{3 1 x^8}{5760}+O\left(x^9\right)$$

  • $\begingroup$ OK, got it, thanks - but calculating the derivatives of e^sinx just gives me longer and longer formulae - how can I simplify it and get the radius of convergence? $\endgroup$ – Simon Mar 8 '14 at 15:55
  • $\begingroup$ Did you just write in $\LaTeX$? Congrats! $\endgroup$ – Git Gud Mar 9 '14 at 15:32
  • $\begingroup$ @GitGud. Most of it is done by my wife ! The remaining is done when I can generate the TeX format ! Cheers. $\endgroup$ – Claude Leibovici Mar 9 '14 at 15:45

For range of convergence, you may want to check out my working at Range of convergence for Taylor's series (about 0) for e^(sin x). Not sure if I did anything wrong though.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.