View attachment 19885
this is written in portuguese but basically I can't solve the first one i) where they ask to show that f(x) (given above) it's the same as the summation given bellow.
Please if someone could help!
First even if the PO were in perfect English, I am not sure what is expected.
I noticed right off that \(f(x)=\dfrac{1}{e^{(x-1)^2}}\)
Let use \(\exp((x-1)^2)=e^{(x-1)^2}\) for economy of notation.
So \(\exp (x) =\displaystyle \sum\limits_{n = 0}^\infty {\dfrac{{{x^n}}}{{n!}}} \) Thus \(\exp (-(x-1)^2) =\displaystyle \sum\limits_{n = 0}^\infty {\dfrac{{{(-1)^n}}}{{n!}}}(x-1)^{2n} \)
You haven't yet told us what you have learned that you can use. I think you said you haven't learn about Maclaurin series. What series do you know?
That's what pka used; see here: https://en.wikipedia.org/wiki/Taylor_series#List_of_Maclaurin_series_of_some_common_functions
The truth is that theoretically I have learned about Maclaurin series, but I can't apply or identify them when solving problems.