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Home Infinite Series Properties of Infinite Series Theorem 2: Tail Of a Series  
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Theorem 2: Tail Of a Series
The next theorem corresponds to the Addition Property for integrals, DEFINITION The series is defined as where b_{n} = a_{m+n}. This series is called a tail of the original series
THEOREM 2 A series
converges if and only if its tail
converges for any m. The sum of a convergent series is equal to the mth partial sum plus the remaining tail, or a_{1 }+ ... + a_{n} + ... = (a_{1} + ... + a_{m}) + (a_{m+1} + ... + a_{m+n} + ...). PROOF First assume the tail converges. For any infinite H, we have a_{1} + ... + a_{H} = (a_{1} + ... + a_{m}) + (a_{m+1} + ... + a_{H}), or Taking standard parts, Therefore the series converges and If we assume the series converges we can prove the tail converges in a similar way.


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