Why is MATLAB not giving me the correct Maclaurin polynomial?

I am trying to evaluate the 4th degree Maclaurin polynomial of the function below using MATLAB:

The following code is the one I used in ,

But clearly this is not the same answer to the one I computed manually by hand. I obtained the 4th degree Maclaurin Polynomial to be:

Answer 1

[John D'Errico]

Lets look at what you did.

If you view this as the exponential function, applied to x^2, thus exp(x^2), we can use the standard Taylor series. That is,

exp(u) = 1 + u + u^2/2 + u^3/6 + u^4/24 + ...

Now, substitute u=x^2, and we have

exp(x^2) = 1 + x^2 + x^4/2 + x^6/6 + x^8/24 + ...

It looks like what you expected. So, you would be correct. Except that you need to read the help for taylor.

What happens in MATLAB?

syms x
taylor(exp(x^2),'order',4)
ans =
x^2 + 1

taylor(exp(x^2),'order',5)
ans =
x^4/2 + x^2 + 1

In the help for taylor, we see:

    'Order'          Compute the Taylor polynomial approximation with
                     order n-1, where n has to be a positive integer. The
                     default value n=6 is used.

So you passed in n=4. It computed a Taylor series approximation of order 3. Yes, I know, the words order and degree are sometimes used as synonyms in mathematics, but then sometimes we see order=degree+1.

[William Choo]

Thank you very much!