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 May 19th, 2017, 05:40 AM #1 Senior Member   Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2 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: 7,638 Thanks: 2623 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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Quote:
 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: 20,476 Thanks: 2039 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: 7,638 Thanks: 2623 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 ?

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