Can You Solve This Infinite Fraction?

What is the value of x= 1/(1 + 1/(1 + 1/(1 + ...)))? This continued infinite fraction has an interesting value!

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[ZachDaChampion]

I just realized that's not your real hand

[dramawind]

When he erased the denominator, he left a tiny bit of the blue circle's top right corner. And that annoyed the crap out of me.

[Tumbolisu]

If you replace every "1+" with "2+", you get the "Silver Ratio". [1+sqrt(2)]
"3+" is the "Bronze Ratio". [(3+sqrt(13))/2]
(n+sqrt(4+n^2))/2
All ratios made this way are called "Metallic Means".

[scottrobinson]

In the comments:
People not understanding how to multiply through by x

[snoW_]

There are a lot of comments asking why the negative solution is disregarded. Remember that the infinite fraction only has 1 value, it is not a solution to an equation, it's an infinite fraction. We transform it into an equation so we can find out what that value is, but we need to have only one solution. Otherwise one could argue that (1-√5)/2 = (1+√5)/2, which is nonsense.

[ArnabAnimeshDas]

When dealing with infinite patterns you can't just ignore the negative term just like that. You have to provide a convincing proof as to why the negative value is not possible.

[georgekousouris]

Had this as my interview question... This video would have been of great use a couple of months ago!!

[gabrielfoos]

You could also create a series un+1=1+1/un then knowing that the associated function 1+1/x is a descending function you know the extracted series u2n and u2n+1 are monotonous and by studying them you see that one is descending and the other is growing. you can use the adjacent series theorem and solve 1+1/l=l to know their limit thus knowing the limit of un that is the golden ratio.This might be a harder way to solve but I just find it fun

[tryhardofdoom]

Never knew one could solve infinite series like this/with this approach :D
I feel dumb not finding the solution on my own since I know recursion from programming :(
But well atleast I learned something new then :D

[C_Lavi]

Thanks alot.. I was struggling with such a question... And I hv the test today only..

[anandkulkarni]

Here is an interesting clue with 6.digit precision for.floats the recursive fraction calculation converges in 32 iterations.

[SantoshKumar-wsqz]

Thank you so much bro. It was very helpful.

[RubyCheetahCub]

How about a fraction that starts with a 1. Next to the one, you put a 1/2, and you do that for every "1" in the fraction. Next to every 2, you put a 2/3 and next to every 3, you put a 3/4, and so on. What is the value of this infinite fraction? (This would probably be more easily explained in a video or picture)

[edwardbartolo]

Inspecting the expression this becomes 1 + 1/x = x. The trick of the author is presenting the expression in purely numerical form. Using algebraic representation immediately shows the expression is repeating itself in the denominator. Let us say, the epression evaluates to x, so, the expression can be written as 1 + 1/entire_infinite_expression = x. But we know entire_infinite_expression is x. So, we have 1 + 1/x = x
==> x -1 = 1/x
==> x(x - 1) = 1
==> x^2 - x - 1 = 0
==> x = { 1 +/-sqrt(1 + 4)}/2
==> x = (1 + sqrt(5))/2
==> x = (1 - sqrt(5))/2

The second aswer is impossible, since all terms are positive. This unwanted root, is the result of multiplying both sides by x.

[TheDavidlloydjones]

Presh,

You are really good at missing the point: at 1.05 you say, without any justification at all "since x only has positive terms...." and decide only to accept (1+sqrt(5))/2 as a solution.

You got the whole thing from assuming that the infinite tower in the original denominator was equal to x, which is fair enough as far as anything having to do with infinities goes. There is nothing in that step that makes x necessarily positive.

This makes you miss the interesting and valuable insight that using the negative value gets you .618033... the reciprocal of the Golden Ratio!

You have foolishly missed one of the basic facts about the Golden Ratio, and one of the most amazing things to be gotten out of this astonishing infinite root tower.

-dlj.

[zigaudrey]

A book give the awnser and this lead to the golden ratio.
1:00 THE GOLD RATIO! YESH!

[emmanuelontiveros]

The Fibonacci numbers also have an relationship with the golden ratio. F(n+1)/F(n)= a, so F(n) comes before F(n+1) in the sequence and a= golden ratio.

[siddharthmehta]

I solved orally
Every indian preparing for jee finds it easy
Very easy

[giorgospapadopoulos]

Today I was thinking that I don't remember you mentioning golden ratio so far.

[Neoncrisisevangelion]

Hey thanks it is in my course of class 10
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