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daniel_i_l
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Homework Statement
Prove that if [tex]0 < \alpha < 1 [/tex] and
[tex]\sqrt[n]{|a_n|} <= 1 - \frac{1}{n^{\alpha}}[/tex] for all n >= 1 then the series
[tex]\sum a_n[/tex] converges.
Homework Equations
The Attempt at a Solution
First I did:
[tex]|a_n| <= \left({1 - \frac{1}{n^{\alpha}}}\right)^n =
\left({{1 - \frac{1}{n^{\alpha}}}^{n^{\alpha}}}\right)^{n^{1-\alpha}}[/tex]
and since the limit at infinity of [tex]\left({{1-\frac{1}{n^{\alpha}}}^{n^{\alpha}}}\right)[/tex] is
[tex]\frac{1}{e}[/tex] then there exists an N so that for all n>N
[tex]|a_n| <= \left({\frac{2}{e}}\right)^{n^{1-\alpha}}[/tex]
But how can I show that [tex] \sum \left({\frac{2}{e}}\right)^{n^{1-\alpha}}[/tex]
converges?
Is there a better way to do this?
Thanks.
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