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sequence convergence


triclino
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Personally - and with no proof or basis - intuitively I think all specified cases converge towards 1. The way the sequence ebbs and flows I cannot envisage any two starting points going off to infinity - but (without maths that is beyond me) there is always a possibility that two starters end up with an oscillating pattern that never calms down

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Personally - and with no proof or basis - intuitively I think all specified cases converge towards 1. The way the sequence ebbs and flows I cannot envisage any two starting points going off to infinity - but (without maths that is beyond me) there is always a possibility that two starters end up with an oscillating pattern that never calms down

 

It is pretty easy to see that if the limit exists then the limit must be 1.

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What I always did with these kind of problems is take that xk is equal to xk+n or xk-n when k goes to infinity,and n doesn't.

 

if xk is equal to some x as x goes to k,than from your equation we get: [math]x=\frac{2}{x+x}[/math]

[math] x=\frac{1}{x}[/math]

[math]x^2=1[/math]

[math]x=1[/math]

 

you can also use the Ratio test which states that [math]x_n[/math] converes if [math]\lim_{n\to \infty}\ \frac{x_{n+1}}{x_n}<1[/math]

Edited by anonimnystefy
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