Abstract
An infinite series En=, tn with partial sums T. is said to be summed to T by the sequence to sequence transformation A = (aij) in case limi, Z= 1 aijTj= T. It is said to be summed to T by the series to sequence transformation B = (bij) in case limi,. j= b 1j tj = T. A sequence to sequence or series to sequence transformation is regular if it sums every convergent series to its ordinary sum. It is well known that corresponding to every regular sequence to sequence transformation A = (a,j) it is possible to construct a regular series to sequence transformation B = (b^^) by defining bij= En== ain; moreover a series with bounded partial sums is summed by A to T if and only if it is summed by B to T. The requirement that the series have bounded partial sums is essential, as is shown by the following example:
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