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January 4th, 2016, 01:57 PM   #1
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Solve this Recurrence Relation?

Can Anyone provide a function satisfying:

f(1) = 1
f(x) = 1 + x/f(x)

This isn't homework or anything, just something I stumbled across.
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January 4th, 2016, 04:14 PM   #2
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Did you type the equations correctly? They don't specify a recurrence relation.

The second equation implies f(1) = 1 + 1/f(1), which isn't true if f(1) = 1.
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January 4th, 2016, 06:11 PM   #3
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Wow, my bad. Here is the correct version:

f(1) = 1
f(x) = 1 + x/f(x-1) for x > 1

Does that make more sense?
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January 4th, 2016, 07:33 PM   #4
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Is f(x) defined only when x is a positive integer?
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January 4th, 2016, 07:37 PM   #5
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Yes
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January 4th, 2016, 10:28 PM   #6
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How did you "stumble across" this?
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January 5th, 2016, 03:37 AM   #7
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Perhaps you mean:

$\displaystyle x_{n+1} = 1 + \frac{n}{x_n}$
$\displaystyle x_1 = 1$

?
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January 5th, 2016, 04:04 AM   #8
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Assuming the iteration scheme is as outlined by my previous post, I calculated the first 1000 values of the function (in Python) and it appears to be very slowly diverging as n increases. Here's a plot of the first 100:
Attached Images
File Type: jpg newval.jpg (7.4 KB, 5 views)

Last edited by Benit13; January 5th, 2016 at 04:08 AM.
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January 5th, 2016, 05:51 AM   #9
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For very large values of $n$, $x_n$ is approximately 1/2 + √$n$.
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