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.© BrainMass Inc. brainmass.com March 4, 2021, 10:05 pm ad1c9bdddf
Determine the uniform convergence and convergence of the series where
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.
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)
Let us first calculate the point wise limit ...
This post deals with uniform convergence and proofs.