Answer
Questioner:   Liva
Category:  Advanced Math
Private:  No
 
Subject:  function_sum_formula
Question:  

functionHello!
I don't know how I'm supposed to make sum formula from function:
f(x)=e(^x^2)+1)  X0=0(I added in attachment the formula in normal way because this way it seemed lame for me)
How much I understand I'm supposed to make derivatives and than sum them. So far it is ok. But how I understand I have to work out general formula... and that is the problem. I know it is somehow connected with MCLoren's (Tailor's special occasion, when x0=0) row (I hope I spelled it correctly. The professor mentioned it, when I asked.)
With hope for some peace of advice, Liva
PS sorry for my English  
.......................
Hi, Liva,

Your idea is good -- write:

e^(x^2 + 1) = e * e^(x^2)

Now you can look up the basic Maclauren series for e^t:
(find it in any calculus book)

e^t = 1 + t + t^2/2! + t^3/3! + ...

or, in summation form:

e^t = SUM[k=0 to inf] t^k/k!

Now do two things:

A. Put t = x^2, and you have:

e^(x^2) = SUM[..] x^2k/k!

B. Multiply each term by  e:

e e^(x^2) = SUM[..] ex^2k/k!

e^(x^2 + 1) = SUM[..] ex^2k/k!

And that is your series.  It looks like:

e + ex^2 + ex^4/2! + ex^6/3! + ...

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