complex analysis

Question

The radius of convergence the series  

  _∝

  ∑  z²ⁿ / zⁿ   is

 ⁿ⁼¹

 

a) 1          b) √2

 

c) √3         c) √5

 

ans is √2 ,but i cant understand how does it happen,  please help me please, i m totally confused, please explain me in details, please sir please

Answer 1

I believe the series you intended to analyze is ∑[n=1,∞] z²ⁿ/2ⁿ. This can written as the power series ∑ a_mz^m where a_m = 2^-m/2 if m is even and 0 otherwise. Then |a_m|^1/m = 2^-1/2 if m is even and 0 otherwise. Hence

 

lim sup |a_m|^1/m = 2^-1/2,

 

and so the radius of convergence is 

 

1/lim sup |a_m|^1/m = 2^1/2,

 

which is answer b).

Answer 2

The sum simplifies to ∑ zⁿ for n=1,...,∞.

 

Use the ratio test

 

a_n = zⁿ so |a_n+1/a_n| = |zⁿ⁺¹/zⁿ| = |z|

 

Recall that if lim_n→∞|a_n+1/a_n| < 1, the series converges. If it's >1 the series diverges. Inconclusive if it's =1

 

Thus your series diverges for |z| > 1 and converges for z < 1.

 

The radius of convergence is therefore 1 or choice (a).

 

Bonus, it can be shown that this series diverges for all complex z where |z|=1 so the convergence set is the open unit disk centered at z=0.