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The terms of the series in the following theorems are supposed to be either real or complex numbers.
Theorem 1 If the series
 and
converge and have the sums  and  , respectively, then also the series
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(1) |
converges and has the sum  .
Proof. The
partial sum of (1) has the limit
Theorem 2 If the series
 converges having the sum  and if  is any constant, then also the series
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(2) |
converges and has the sum  .
Proof. The
partial sum of (2) has the limit
Theorem 3 If the terms of any converging series
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(3) |
are grouped arbitrarily without changing their order, then the resulting series
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(4) |
also converges and its sum equals to the sum of (3).
Proof. Since all the partial sums of (4) are simultaneously partial sums of (3), they have as limit the sum of the series (3).
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