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help me please? i desperately need help with my math?
find the first 4 terms i the MacLaurin Series for e^-x2
how to solve it? what should we do first?
1 Answer
- chauncyLv 71 decade agoFavourite answer
I assume you mean e^(- x^2). Did you want the first 4 terms including or excluding zeroes?
The Maclaurin series expansion for f(x) is
f(x) = f(0) + x f ' (0) + x^2/2! f ' ' (0) + ...
If f(x) = e^(- x^2) then
f ' (x) = - 2x e^(- x^2)
and f ' (0) = 0
f ' ' (x) = (4x^2 - 2) e^(- x^2)
and f ' ' (0) = - 2
f ' ' ' (x) = (- 8x^3 + 12x) e^(- x^2)
and f ' ' ' (0) = 0
So the Maclaurin series expansion for e^(- x^2) is
e^(- x^2) = 1 + 0 - x^2 + 0 ...
For further terms, just continue the process above.
P.S. It has just occurred to me that there is a simpler way.
Let - x^2 = y and do the expansion for g(y) = e^y as follows:
g(y) = g(0) + y g ' (0) + y^2/2! g ' ' (0) + ...
= 1 + y + y^2/2! + y^3/3! + ...
Then substitute - x^2 for y as follows:
e^(- x^2) = 1 - x^2 + x^4/2 - x^6/6 + ...