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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 sin⁡nx ) is uniformly convergent, then lim⁡(nk_n )=0.

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    https://brainmass.com/math/real-analysis/uniform-convergence-proof-297769

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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 ...

    Solution Summary

    This post deals with uniform convergence and proofs.

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