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# Prove some properties of the function e-1/x2

Consider the function

and .

1. Prove that for every positive number we have

2. Prove that if then

where is a polynomial in .

3. Prove that

1. Proof. (A specific case of this general theorem is actually the first problem of this section, here. Maybe it’s worth taking a look since this proof is just generalizing that particular case.) We make the substitution , so that as and we have

by Theorem 7.11 (page 301 of Apostol) since implies as well

2. Proof. The proof is by induction on . In the case we have

So, indeed the formula is valid in the case . Assume then that the formula holds for some positive integer . We want to show this implies the formula holds for the case .

is still a polynomial in since the derivative of a polynomial in is still a polynomial in , and so is the sum of two polynomials in . Therefore, we have that the formula holds for the case ; hence, it holds for all positive integers

3. Proof. The proof is by induction on . If then we use the limit definition of the derivative to compute the derivative at 0,

So, indeed and the statement is true for the case . Assume then that for some positive integer . Then, we use the limit definition of the derivative again to compute the derivative ,

This follow since is still a polynomial in , and by the definition of for . But then, by part (a) we know

Therefore,

Thus, the formula holds for the case , and hence, for all positive integers