Prove the enclosed area of this function is equal to 1

[Saracen Rue]

Homework Statement

If function $f$ is defined as such that $f\left(x\right)=x^{\frac{1}{\sum_{n=1}^∞x^n}}$, then prove that the area enclosed between the the derivative function, $f'(x)$, and the $x$-axis is equal to $1$ sq unit

Homework Equations

Knowing that the area under a function $f(x)$ between two can be found by using $\int_b^a\left|f\left(x\right)\right|dx$

The Attempt at a Solution

I'm having quite a lot of trouble with this question. I know that the first step should be to try and find the points at which the function intercepts the x-axis but I'm unsure of how to do this when the function has an infinite sum in it. Any help with this will be greatly appreciated :)

 

[Metmann]

You should first rewrite the exponent. What is $\sum_{n=1}^{\infty} x^n$ for $|x|>= 1$? For $|x|<1$ you should recognize the geometric series (but be careful: The first term (n=0) is missing here). After rewriting the exponent, you obtain a much simpler x-dependence.

(Result should be: $f(x) = x^{\frac{1-x}{x}}$ for $|x|<1$ and $f(x) \equiv 1$ for $|x|>=1$. In fact your function is not defined for $x=-1$ but you could define it as $1$ at this point, because it converges from left and right to $1$)

 

[fresh_42]

Are negative values of $x$ allowed? The function is pretty crazy for them between $-1 < x < 0$. For real positive values only, the fundamental theorem of calculus should do.

 

[Metmann]

The function is pretty crazy for them between $-1 < x < 0$.

Indeed, it should even become complex, right?

 

[fresh_42]

Yes.