# 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