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If a sum of a finte number of terms is infinite, does that imply that at least one term in the finite sum is also infinite?

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Yeah, because if the terms were all finite, their finite sum would be finite. This is the contrapositive of your statement. –  Antonio Vargas Dec 10 '12 at 20:36

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Consider $A = a_{1} + \cdots + a_{n}$. If $a_{1} , \ldots , a_{n}$ are all finite, define $$ \alpha = \max(|a_{1}| , \ldots , |a_{n}|). $$ Then, we have that $$ |A| = |a_{1} + \cdots + a_{n}| \leq |a_{1}| + \cdots + |a_{n}| \leq n \alpha, $$ so that $A$ is also finite.

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