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Abel's Lemma Abel's test Abel's theorem absolutely convergent apply asymptotic series binomial series calculation coefficients complex condition consider constant continuous function convergent series converges absolutely converges uniformly corresponding curve deduce definite limit denote differential equation Dirichlet's Dirichlet's test divergent series double series easy equal Euler's example expansion exponential expression follows formula fraction gives greater Hence increases inequality integer integral interval ISBN logarithmic Math method Napier's negative notation number of terms numerically less obtain oscillates polynomial positive integer positive number positive terms power-series Pringsheim proof prove radius of convergence rational numbers reader result satisfied sequence series converges shew Similarly Sn(x solution steadily decreases suppose tends steadily tends to infinity tends to oo tends to zero term-by-term theorem Art theory uniform convergence upper limit variable verify Weierstrass's write