## A Geometrical Method for the Treatment of Uniform Convergence and Certain Double Limits |

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AMERICAN MATHEMATICAL SOCIETY arith arithmetic form arithmetic proof belt calculus chosen at pleasure conditionally convergent continuous derivative continuous func continuous function sn(x continuous limit convergence of Sn(x converges absolutely corresponding series curve y d f(x y differentiated term differentiating a series Double Integral double limit means GEOMETRICAL METHOD Hankel's hence higher approximation curves independent variables indicated in Fig integrated term integrating a series last expression Let sn(x Let the series lim sn(x limiting curve limiting locus locus y=f(x main question mation curves metic form non-uniform order of integration peaks Plot the approximation positive quantity power series reverse the order rn(x series a0 series of continuous series term sign of integration sm(x Stolz strip sufficient condition telescope series term by term theorem throughout the region tinuous functions tion true equation u-series uniform convergence uniformly convergent series ut(x value of x variable is true W. F. OSGOOD y)dy