Let the real sequence x n be given by, ∑ j = 1 2 n 1 j − ∑ j = 1 n 1 j .

Question

Let the real sequence             x      n       be given by,      ∑          j      =      1              2      n            1    j    −      ∑          j      =      1              n            1    j    .

Terrance Phillips · Accepted Answer

First note that the sequence   (      x    n    ) is bounded above. This follows from your observation that       ∑          j      =      1              n            1          j      +      n      . Here we have n terms, all of them clearly less than   1      /    n, so their sum is less than 1.Next you want to show that the sequence   (      x    n    ) is increasing. Calculate       x          n      +      1        −      x    n  , and show it is positive. Most of the terms cancel:      x          n      +      1        −      x    n    =      1          2      n      +      1        +      1          2      n      +      2        −      1          n      +      1        .Finally, appeal to the theorem that an increasing sequence which is bounded above has a limit.

Let the real sequence <mrow class="MJX-TeXAtom-ORD"> x n </msub> </mrow>

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