\[\large{\alpha(n)=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+....+\frac{1}{2^{n}-1}}\]

For a positive integer \(n\). define \( \alpha(n) \) as above, then which of these are true?

\(A)\quad\) \(\alpha(100)\leq 100\)

\(B)\quad \) \(\alpha(100) >100\)

\(C)\quad\) \(\alpha(200)\leq 100\)

\(D)\quad \) \(\alpha(200)>100\)

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