Friday, 16 June 2017

sequences and series - Show that an defined by an+1=fracan+2an+1 converges

Suppose a0 is an arbitrary positive real number. Define the sequence {an} by an+1=an+2an+1

for all n0. I have to prove that {an} converges.



My attempt: If a=limnan exists, then it should be a solution to a=a+2a+1

which is 2. Thus I need to show that |2an| gets arbitrarily small for large n. I tried to prove that |2an|<|2an1| but couldn't.

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