Abel theorem

Abel theorem [′ä·bəl ′thir·əm]

(mathematics)

A theorem stating that if a power series inzconverges forz=a, it converges absolutely for |z| < |a|.

A theorem stating that if a power series inzconverges tof(z) for |z| < 1 and toaforz= 1, then the limit off(z) aszapproaches 1 equalsa.

A theorem stating that if the three series withnth terma_n,b_n, andc_n=a₀b_n+a₁b_n-1+ ⋯ +a_nb₀, respectively, converge, then the third series equals the product of the first two series.