May 19th, 2017, 05:40 AM  #1 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  Recurrence Sequence
The nonzero 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  
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  Quote:
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 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  
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 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  

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