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?
Finite sums of infinite value
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sequences-and-series
infinity
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0Yeah, because if the terms were all finite, their finite sum would be finite. This is the contrapositive of your statement. – 2012-12-10
1 Answers
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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.