Real Analysis I am proving that the telescopic series (a1-a2)+(a2-a3)+. . . converges IFF the sequence {an}n=1 to infinity. I posted this question and got a response. I mostly understand it but have a few questions. I am told to have Sn=Sum (ai-ai+1) from i=1 to n. Do I call Sn the sequence of partial sums? Because if the sequence of partial sums is convergent then the series converges. If Sn is not the sequence of partial sums, then what is it? And do I say "Let Sn= Sum . . . " In another part of the answer I was given, when we suppose that the sequence {an} converges we arrive at Sn=a1-an+1 Then we take the limit of both sides. I understand why the limit of the RHS exists but why does the limit of the LHS (Sn) exist? Also, when I have proven (2)sequence converges so series converges, how do I state the conclusion? Do I say that since Snis the sequence of partial sums of the series and Sn converges then the Sum (ai-ai+1) from i=1 to n converges and thus (a1-a2)+(a2-a3)+ . . is convergent? Is that the appropriate wording?
I am proving that the telescopic series (a1-a2)+(a2-a3)+. . . converges IFF the sequence {an}n=1 to infinity.
I posted this question and got a response. I mostly understand it but have a few questions.
I am told to have Sn=Sum (ai-ai+1) from i=1 to n. Do I call Sn the sequence of partial sums? Because if the sequence of partial sums is convergent then the series converges. If Sn is not the sequence of partial sums, then what is it? And do I say "Let Sn= Sum . . . "
In another part of the answer I was given, when we suppose that the sequence {an} converges we arrive at
Sn=a1-an+1
Then we take the limit of both sides. I understand why the limit of the RHS exists but why does the limit of the LHS (Sn) exist?
Also, when I have proven (2)sequence converges so series converges, how do I state the conclusion? Do I say that since Snis the sequence of partial sums of the series and Sn converges then the Sum (ai-ai+1) from i=1 to n converges and thus (a1-a2)+(a2-a3)+ . . is convergent? Is that the appropriate wording?
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