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where the partial sums BN = b0 + ··· + bn are bounded, λn has bounded variation, and lim λn Bn exists:
then the series ∑ an is convergent. This applies to the pointwise convergence of many trigonometric series, as in
with 0 < x < 2π. Abel's method consists in writing bn+1 = Bn+1 − Bn, and in performing a transformation similar to integration by parts (called summation by parts), that relates the given series ∑ an to the absolutely convergent series
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