Given the sequence #a_1=sqrt(y),a_2=sqrt(y+sqrt(y)), a_3 = sqrt(y+sqrt(y+sqrt(y))), cdots# determine the convergence radius of #sum_(k=1)^oo a_k x^k# ?
1 Answer
If
Explanation:
Assuming we want to deal with Real numbers only, we require
If
Otherwise,
Note that the sequence
It does have a finite fixed point towards which it converges:
Let
Then:
#t^2-t-y = 0#
So using the quadratic formula:
#t = (1+-sqrt(1+4y))/2#
and since
#t = 1/2+sqrt(1+4y)/2 = 1/2+sqrt(y+1/4)#
This is a fixed point of the function
In particular, if
#sqrt(y) sum_(k=1)^oo x^k <= sum_(k=1)^oo a_k x^k <= (1/2+sqrt(y+1/4)) sum_(k=1)^oo x^k#
So
Hence the radius of convergence is