Answer to Question #212145 in Real Analysis for Nikhil

Question #212145

Show that series Σ x/(1+n^2.x^2) is uniformly convergent in [k,1] where k>1 but not uniformly convergent in [0,1]


1
Expert's answer
2021-07-02T10:24:40-0400

If "x\\in[k,1]" , "0<k<1" , then "\\frac{x}{1+n^2x^2}\\leq \\frac{x}{n^2x^2}=\\frac{1}{n^2x}\\leq \\frac{1}{n^2 k}".

The expression on the right hand does not depend on x and the series "\\sum\\limits_{n=1}^{\\infty}\\frac{1}{n^2 k}" is convergent, therefore the initial series is uniformly convergent on the segment [k,1].


In order to prove that the series is not uniformly convergent on the segment [0,1], it is sufficient to show that for any "N\\in\\mathbb{N}" there exist "x\\in[0,1]" , "q> p> N" such that "\\sum\\limits_{n=p}^{q}\\frac{x}{1+n^2x^2}\\geq 1\/5" .

Put x=1/N, p=N+1, q=2N. Then "nx\\leq 2" for all "n=p, p+1, \\dots, q" and

"\\sum\\limits_{n=p}^{q}\\frac{x}{1+n^2x^2}\\geq \\sum\\limits_{n=p}^{q}\\frac{x}{1+2^2}=\\sum\\limits_{n=N+1}^{2N}\\frac{1\/N}{5}=1\/5"

Therefore, the series "\\sum\\limits_{n=1}^{\\infty}\\frac{x}{1+n^2x^2}" is not uniformly convergent on the segment [0,1].


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