Saturday, 20 August 2016

Convergent Sequence Terminology



What is the following sequence classified as? I don't want to make anybody solve it, I just need to know where to begin looking to solve it.
α1=20


αn+1=20+αn



I am suppose to prove that it converges to 5, however if I could just get a little terminology help it is more then appreciated!




Note: I updated the terminology, as well as give the initial value.



Thanks!


Answer



First, it's not a series, it's a sequence. Fixed in the original.



Second, it's a recursively defined sequence.



A sequence is "recursively defined" if you specify some specific values and then you explain how to get the "next value" from the previous one; much like induction. Here, you are saying how to get the "next term", αn+1, if you already know the value of the nth term, αn.




Once you know the first value, then the sequence is completely determined by that first value and the "recurrence rule" αn+1=20+αn.



Now some hints:




  • Show the sequence is increasing.

  • Show the sequence is bounded.

  • Conclude the sequence converges.

  • Once you know it converges, take limits on both sides of the recursion to try to figure out what it converges to.



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