Difficult Calculus Problem Below. Can anyone attempt to solve and please explain?

Question

What is the largest open interval of convergence (don't worry about endpoints) for the power series about c = 0 which represents the function f(x) = 2/(x^2 + 4) ?

don't worry about endpoints...

possible book solutions

a. (-1/6, 1/6)

b. (-1/4, 1/4)

c. (-1/3, 1/3)

d. (-1/2, 1/2)

e. (-1, 1)

f. (-3/2, 3/2)

g. (-2, 2)

h. (-3, 3)

i. (-4, 4)

or is it none of these?

Answer 1

g: (-2, 2)

All power series converge on the entire interior of a disk centered at the point of expansion with a radius r which is called (unsurprisingly) the radius of convergence. This radius is always the distance from the center of the disk to the nearest singularity on the complex plane. In this case, the function f(x) has exactly two singularities on the complex plane, namely x=2i and x=-2i. The distance from 0 to both of these singularities is 2 and so the radius of convergence is also 2. So on the real line, it converges for all values between -2 and 2.

Answer 2

Looks like none of these to me. If I remember correctly, this function looks like it converges at zero (0) only. As x approaches infinity, f(x) converges on zero.

Answer 3

P) As n techniques infinity, (4^n -a million)/(3^n + a million) techniques (4/3)^n. because 4/3 is extra suitable than one, it gets larger and better with out ever drawing near any variety. subsequently, it diverges. Q) -ln(n) = ln(a million/n). So to establish that Q to be genuine, secx + tanx might desire to be equivalent to a million/(secx - tanx). secx - tanx = a million/cosx - sinx/cosx = (a million-sinx)/cosx. a million/(all that relaxing stuff) is cosx/(a million - sinx), it is cosx - cotx. that isn't in any way equivalent to secx + tanx, so this fact isn't genuine. subsequently, purely P is genuine.