For every n =1,2,3,... and x >=0, let f_n(x) = (1^x +2^x....+ n^x)/(n^(x+1)). Then, does f_n converge to some function f?
If x is an integer, then, by means of those expressions for the sum of integer powers of the natural numbers, it's easy to show lim f_n(x) = 1/(x+1). But does this remain true for every x >0?
If f_n converges to some f, then is the convergence uniform?
If we differentiate each f_n, then does f'_n converge fo f', supposing f_n --> f?
2007-09-11
04:14:45
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3 answers
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asked by
Sonia
1
in
Mathematics