how would you determine if if the series converges absolutely, conditionally or diverges? series n=1 to infinity (-5)^-n
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Let $\displaystyle a_n = (-5)^{-n} = (-1)^{-n}5^{-n}$. Then $\displaystyle |a_n| = 5^{-n}$. Now, notice that: $\displaystyle \sum |a_n| = \sum 5^{-n} = \sum \left(\frac{1}{5}\right)^n$ which is a geometric series.
Originally Posted by o_O Let $\displaystyle a_n = (-5)^{-n} = (-1)^{-n}5^{-n}$. Then $\displaystyle |a_n| = 5^{-n}$. Now, notice that: $\displaystyle \sum |a_n| = \sum 5^{-n} = \sum \left(\frac{1}{5}\right)^n$ which is a geometric series. Just to add $\displaystyle \sum_{n=1}^\infty (-5)^{-n} = \sum_{n=1}^\infty \left(\frac{-1}{5}\right)^n$ is also geometric.
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