How can the Taylor expansion of x^x at x=1 be simplified to make solving easier?

[nezahualcoyot]

Homework Statement

Find the Taylor expansion up to four order of x^x around x=1.

Homework Equations

The Attempt at a Solution

I first tried doing this by brute force (evaluating f(1), f'(1), f''(1), etc.), but this become too cumbersome after the first derivative. I then tried writing: $$x^x = e^{x \ln(x)}$$

And found the Taylor expansion of x*ln(x) (which I can do), and the "plug" that into the Taylor expansion of e^x, and carefully only keep the terms up to four order. I checked the final result with Wolfram Alpha and I got it correct, but this procedure took me way too long (specially the last step) and feels way harder than the rest of my course.

My question is, is there an alternative / easier way of solving this problem?

 

[lekh2003]

You seemed to have found an easy way to do it without brute force. In my opinion, you've used quite an efficient way using the fact that $e^x$ is its own derivative.

But await on some more answers from blokes with experience in Calculus. I've only just started learning...

 

[vela]

The way you solved it seems pretty straightforward. Of course, a lot of it depends on how you did the algebra. You can end up doing a lot of unnecessary work.