Consider the power series f(x)=∑ k=0
[infinity]
​
5k−1
2 k
​
(x−1) k
. We want to determine the radius and interval of convergence for this power series. First, we use the Ratio Test to determine the radius of convergence. To do this, we'll think of the power series as a sum of functions of x by writing: ∑ k=0
[infinity]
​
5k−1
2 k
​
(x−1) k
=∑ k=0
[infinity]
​
b k
​
(x) We need to determine the limit L(x)=lim k→[infinity]
​
∣
∣
​
b k
​
(x)
b k+1
​
(x)
​
∣
∣
​
, where we have explicitly indicated here that this limit likely depends on the x-value we choose. We calculate b k+1
​
(x)= and b k
​
(x)= Exercise. Simplifying the ratio ∣
∣
​
b k
​
b k+1
​
​
∣
∣
​
gives us ∣
∣
​
b k
​
b k+1
​
​
∣
∣
​
=∣ ∣x−1∣