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 May 19th, 2017, 05:40 AM #1 Senior Member   Joined: Nov 2015 From: hyderabad Posts: 242 Thanks: 4 Recurrence Sequence The non-zero values for $x_0$ and $x_1$ such that the sequence defined by the recurrence relation $x_{n+2} = 2x_n$, is convergent are A) $x_0 = 1$ and $x_1 = 1$ B) $\displaystyle x_0 = \frac{1}{2}$ and $\displaystyle x_1 = \frac{1}{4}$ C) $\displaystyle x_0 = \frac{1}{10}$ and $\displaystyle x_1 = \frac{1}{20}$ D) none of the above I have checked the sequence, but I was not able to understand which is monotonically increasing or decreasing to check the convergence. Option A is having the sequence like 1,1,2,2,4,4,8,8... Is it convergent? Please help! Last edited by skipjack; May 19th, 2017 at 08:01 AM. May 19th, 2017, 05:50 AM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,691 Thanks: 2670 Math Focus: Mainly analysis and algebra I would calculate numbers for each of A, B, C and D first. If you need more information after that, look at the definition of convergence: informally a sequence is convergent if it gets closer and closer to some limiting value. Is that true of 1,1,2,2,4,4,8,8? An alternative approach is to consider that every subsequence of a convergent sequence converges to the same limit as the sequence. So, does the subsequence $(x_0, x_2, x_4, \ldots)$ converge? Last edited by v8archie; May 19th, 2017 at 05:52 AM. May 19th, 2017, 06:26 AM   #3
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 Originally Posted by v8archie I would calculate numbers for each of A, B, C and D first. If you need more information after that, look at the definition of convergence: informally a sequence is convergent if it gets closer and closer to some limiting value. Is that true of 1,1,2,2,4,4,8,8? An alternative approach is to consider that every subsequence of a convergent sequence converges to the same limit as the sequence. So, does the subsequence $(x_0, x_2, x_4, \ldots)$ converge?
The subsequence $2^n$ is divergent in Ratio test. Does it mean the original sequence also diverges?

All subsequences of the 3 options are diverging... Does it mean none of the above is the correct option?

Last edited by skipjack; May 19th, 2017 at 08:02 AM. May 19th, 2017, 08:02 AM #4 Global Moderator   Joined: Dec 2006 Posts: 21,036 Thanks: 2273 Yes. May 19th, 2017, 08:04 AM   #5
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 Originally Posted by skipjack Yes.
You are telling YES to my last message or to my first question ?? May 19th, 2017, 08:28 AM #6 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,691 Thanks: 2670 Math Focus: Mainly analysis and algebra Both. Think about it. May 19th, 2017, 08:35 AM   #7
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 Originally Posted by v8archie Both. Think about it.
If a subsequence is divergent then it's original sequence also diverges ? Tags recurrence, sequence Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post randalspanish Math 0 April 13th, 2015 02:34 PM greggor Complex Analysis 2 April 10th, 2015 03:02 AM JSimmonds49 Algebra 1 August 18th, 2013 05:54 PM natkoza Real Analysis 2 December 6th, 2010 01:20 PM Student 100 Computer Science 2 November 25th, 2008 05:52 AM

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