※ 引述《ss1132 (景)》之銘言：
: Discuss the uniform convergence of
: gn(x)=sum(k=1,n,x/(1+k^2x^2)) 0<x<infinity
Suppose gn→g uniformly as n→∞
then there exists N (independent of x) such that
g-gn < 1/10 whenever n≧N
however,
g(x)-gN(x) > sum(k=N+1,2N,x/(1+k^2x^2) > sum(k=N+1,2N,x/(1+4N^2x^2)
take x=1/(2N)
then
g(1/(2N))-gN(1/(2N)) > (1/(2N))(N/2) = 1/4 > 1/10
contradiction
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