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For brevity, notation and terminology in this note are generally not explained. I t is believed that they will be clear in all instances to any probable reader. THEOREM. The Holder and Cesàro methods of summation are equivalent. PROOF. Let Cn (2) represent the Cesàro sum of order r to n terms: r r(n — k + 1) • • • n (1) Cnw = so + • • • + * * r + n (r + n — k) • • • (r + n) r(n\) + • • • +sn —— • r • • • (r + n) We readily verify that Cn (r) satisfies the equation (2) (n + r + 1) C„<**> -nC ^ = (r + 1)C»<'>,

An elementary proof by mathematical induction of the equivalence of the Cesàro and Hölder sum formulas