Determine the uniformconvergence 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

(1) Let G = {z : 0 < abs(z) < R} for some R > 0 and let f be analytic on the punctured
disk G with Laurent Series f(z) = sum a_n*z^n (from n = -oo to oo).
(a) If f_n(z) = sum a_k*z^k (from k =-oo to n), then prove that f_n converges pointwise
f in C(G,C) (all continuous functions from G to C (complex)); i.e., {f_n}

Please see the attached image for complete questions.
1. Find the interval of convergence (including a check of end-points) for each of the given power series.
2. Use the geometric series test (GST) to write each of the given functions as a power series centred at x=a, and state for what values of x the series converges.

1.) Find the interval of convergence of the series Σ (for n=0 to ∞) (4x-3)^(3n)/8^n and, within this interval, the sum of the series as a function of x.
2.) Determine all values for which the series Σ (for n=1 to ∞) (2^n(sin^n(x))/n^2 converges.
3.) Find the interval of convergence of the series Σ

The problem is to determine the radius of convergence of the Taylor Series for each of the functions below centered at x. We are to explain our conclusion in each case. I would like to see how to work each problem (including what the Taylor Series is) and what the explanation is.
a) centered at and
NOTE: I know the

Test for convergence or divergence
1.) sum from n=1 to infinity of (e^1/n)/(n^2)
2.) sum from j=1 to infinity of (-1)^j * ((sqrt j)/(j+5))
3.) sum from n=2 to infinity of (1/((1+n)^(ln n))
keywords: tests