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If $|a_n| < 10^{-n}$, prove that $\sum^{\infty}_{n=1} a_n$ converges.

Could someone give me a hint as to how to start this?

2 Answers 2

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Hint If $|a_n|<10^{-n}$, then $\sum|a_n|<\sum 10^{-n} <\infty$

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    To be sure even more, using the fact that, absolute convergence leads to convergence - we done.2012-04-03
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If $A_n = \sum^{n}_{i=1} a_i$, if $n > m$, $A_n - A_m = \sum^{n}_{i=m+1} a_i < \sum^{n}_{i=m+1} 10^{-i} < 10^{-m}$ (you can do better, but this is enough). Then apply the Cauchy criterion.

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    Then |A_{n} - A_{m}| < 10^{-m}. Since $10^{-m} \to 0$, |A_{n} - A_{m}| < \epsilon for n > m \geq N.2012-04-03