Question

How to prove that `{2^n}` is unbounded real series ?

Answer 1

By definition a real series `{a_n}` is bounded if we can find an `M in RR` and an integer `N` for which:

`n > N => abs(a_n) < M`

For `a_n = 2^n` first we not that all terms are positive, so that:

`abs(a_n) = a_n`

Then we can see that for any `M in RR` if we choose `N > lnM xx ln2` then for `n > N`:

`a_n = 2^n > 2^N > 2^(lnM xx ln2) = (2^ln2)^lnM = e^lnM = M`

Hence for any `M in RR` we can find a term of the series such that:

`abs(a_n) > M`

which proves the series is unbounded.