uniform convergence and proof
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Determine the uniform convergence and convergence of the series ∑▒〖〖(f〗_n),〗 where f_n (x)is given by the following: (The Weoerstrass M-Test will be needed)
a sin(x/n^2 ) b. 〖(nx)〗^(-2),x≠0,
c. 〖(x^2+n^2)〗^(-1) d. (-1)^n (n+x)^(-1),x≥0,
e. 〖(x^n+1)〗^(-1),x≥0 f. x^n (x^n+1)^(-1),x≥0.
Suppose (k_n ) is a decreasing sequence of positive numbers. If ∑▒(k_n sinnx ) is uniformly convergent, then lim(nk_n )=0.
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This post deals with uniform convergence and proofs.
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Determine the uniform convergence and convergence of the series where
a b.
c. d.
e. f.
a)
We know for all x, we have
For any given x, is convergent hence the series is convergent.
Further, as for any x. Hence the convergence is uniform.
b)
Let x is not in (-1, 1). Then we have . Hence the convergence of is uniform.
Let x is in (-1, 1) but not equal to zero. Let us suppose, . Then y is not in (-1, 1). So, we have
For a given such y, we have . Hence the convergence is pointwise.
Now, does not exist. Hence the convergence is not uniform in the interval (-1, 1)
c)
Let us first calculate the point wise limit ...
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