Infinite fraction function (with ascending powers)

[Gatineau]

In picture #1 is everything I've gathered. My question is how can you simplify the function into a more manageable form?
I have no idea what's going on when the function gets negative values of x. It seems all over the place, but I do know f(-1) = 1.

• How to simplify function into a more manageable form?
• What's up with the negative value chart of this function?
• Using the simplified form, find the exact value of the maxima if it (the simplified function) is differentiable.
• Prove the limits using the simplified function.

 

[BigBeachBanana]

This thread would be helpful for you. It's not an exact question, but there are some ideas you can explore.

Derivative of a finite continued fraction

I got this as a challenge question from my professor, but not sure how to start. Let the fraction be be iterated 2018 times, and f(x)=x-\frac{1}{x-\frac{1}{...-\frac{1}{x-1}}} Find f'(0)

www.freemathhelp.com

 

[Gatineau]

Thanks for the link!

I didn't manage to find anything algebraic of worth. I also couldn't find any tools on the internet that could help with this, but I did graph the function again, this time including negatives.
Observations:

• Vertical asymptotes become more and more common the more left you go.
• All x-intercepts and H.A. intercepts are left of the y-axis.
• There appear to be some inflection points(?) between the vertical asymptotes. I'm unsure about where they are if they exist.
• You'll notice that I extended the H.A. because I noticed that as x approaches negative infinity y approaches 1, from the bottom. However, I used a very quick and unsophisticated program I made to check that, so I can't be certain.
• It sort of reminds me of the factorial function..? It, too, eventually approaches a value towards negative infinity, and has some craziness to the left of the y-axis. It also has that "hook" near x=0. Though as x approaches +infinity y does, too, in the factorial function; as opposed to this one.

I'll continue to try to figure it out, and if I find a solution I'll post it here. Algebra-wise, I'm currently trying to find some way to differentiate using something along the lines of what you provided.

 

[BigBeachBanana]

Thanks for the link!

I didn't manage to find anything algebraic of worth. I also couldn't find any tools on the internet that could help with this, but I did graph the function again, this time including negatives.
Observations:

• Vertical asymptotes become more and more common the more left you go.
• All x-intercepts and H.A. intercepts are left of the y-axis.
• There appear to be some inflection points(?) between the vertical asymptotes. I'm unsure about where they are if they exist.
• You'll notice that I extended the H.A. because I noticed that as x approaches negative infinity y approaches 1, from the bottom. However, I used a very quick and unsophisticated program I made to check that, so I can't be certain.
• It sort of reminds me of the factorial function..? It, too, eventually approaches a value towards negative infinity, and has some craziness to the left of the y-axis. It also has that "hook" near x=0. Though as x approaches +infinity y does, too, in the factorial function; as opposed to this one.

I'll continue to try to figure it out, and if I find a solution I'll post it here. Algebra-wise, I'm currently trying to find some way to differentiate using something along the lines of what you provided.
View attachment 32229

I'm not sure what you're trying to accomplish exactly. But you asked how to get it into a more manageable form. Using a recursion would make your equation more manageable. Don't you agree?