My Math Forum Recurrence Sequence

 Real Analysis Real Analysis Math Forum

 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: 7,305 Thanks: 2443 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

Posts: 232
Thanks: 2

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: 18,951 Thanks: 1599 Yes.
May 19th, 2017, 08:04 AM   #5
Senior Member

Joined: Nov 2015

Posts: 232
Thanks: 2

Quote:
 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: 7,305 Thanks: 2443 Math Focus: Mainly analysis and algebra Both. Think about it.
May 19th, 2017, 08:35 AM   #7
Senior Member

Joined: Nov 2015

Posts: 232
Thanks: 2

Quote:
 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 Display Modes Linear 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

 Contact - Home - Forums - Cryptocurrency Forum - Top