Deducing Maclaurin series converges from Leibniz formula

sr3056

Given f(x) = xe^-x2 I can differentiate once and use Leibniz to show that for n greater than 1

f⁽ⁿ⁾ = -2nf^(n-2) - 2xf^(n-1)

I want to show that the Maclaurin series for f(x) converges for all x.

At x = 0, the above Leibniz formula becomes f⁽ⁿ⁾ = -2nf^(n-2)

I know that f(0) = zero so this implies that even terms of Maclaurin series are zero, whilst
f⁽¹⁾(0) = 1, f⁽³⁾(0) = -6*1 = -6, f⁽⁵⁾(0) = -10*-6*1 = 60 and so on.

I assume that to show convergence, I need to find a formula for the nth term of the Maclaurin series, then use the ratio test to show that terms are decreasing. I can see the pattern (the derivative increases by a factor of 2(2n + 1) each time) but am unsure how to express this in a formula. Perhaps using factorials?

Thanks for any help.

chiro

Hey sr3056 and welcome to the forums.

One suggestion is to use the Mclaurin series expansion for e^(-x^2) and then multiply all terms by x. So you start by expanding e^(-x) and then replace every x by an x^2, and multiply each term by x.

The above should give you a series expansion which should allow you to do a further test for convergence.

sr3056

Thanks. Is there no way of proving convergence from the Leibniz formula though?

chiro

If you get a series that is the correct definition, then you should be able to do what you said (i.e. the ratio test), so as long as you get the correct series expansion, it will be OK.

Both series expansions should be equal though and testing this will test whether your above approach is the same as the one I discussed above.