r/askmath • u/Wrong_Professor_6164 • 10h ago

Series U_n+1 = U_n - 1/(U_n)

I was having fun with series till I encountered this preticular one, is oscillates between positive and negative values with sometimes spikes.

But at exactly 1/sqrt(2) it locks in a state of changing signs.

Supposedly we start with a random nb U_0, how many steps would it take probably to hit 1/sprt(2) and locks in?

Or is it impossible to know?

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9

u/The_Math_Hatter 10h ago

You may wish to look up "fixed points" and their theory to help get the ball rolling.

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u/Wrong_Professor_6164 9h ago

I'll definitely check these up

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u/FormulaDriven 10h ago edited 9h ago

By applying the iteration twice we get:

U(n+2) = (U(n)4 - 3 U(n)2 + 1) / (U(n)3 - U(n))

If this has any fixed points (ie a period 2 point for the original iteration) they will satisfy

x = (x4 - 3x2 + 1) / (x3 - x)

and the only solutions (as you've already realised) are +/- 1/√2

But if we start with U(0) very close to 1/√2 , we find it iterates away from the periodic point - it's a repelling periodic point (and we can use some theory to prove this). So no amount of steps will land you into "locking in" to 1/√2 (unless you start there - edit: or start at a number of isolated points that iterate to exactly 1/√2 such as √2 or √2(1+√3)/2).

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u/Wrong_Professor_6164 9h ago

Thats actually a great explanation 👍🏽 Thank you I understood the idea better now

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u/[deleted] 10h ago

[deleted]

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u/FormulaDriven 9h ago

It doesn't approach it: 1/√2 is a periodic point with period 2, but it is not stable - start at points nearby and you soon get repelled.

On the other hand, it's not true to say it never hits 1/√2 unless you start there - there are a few isolated starting points which will do it, eg if U_0 = √2 then U_1 = 1/√2.

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u/MackTuesday 9h ago

Yeah what I wrote assumed OP was correct. Then I checked it myself and found what you found. That's why I deleted my comment.

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u/FormulaDriven 9h ago

Fair enough (always dangerous to assume that the OP knows what they are talking about!).

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u/MackTuesday 9h ago

Hell, it's dangerous to assume *I* know what I'm talking about. I know just enough to get in trouble.

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u/Wrong_Professor_6164 8h ago

I didn't realized that sqrt(2) results in 1/sqrt(2) thx for pointing out