6. (a) Explain why the ratio test will not let us calculate the radius of convergenceof the power series∞Xn=0anxn, where an = 1 : n is a power of 2,2 : n is not a power of 2.[7 marks](b) Use the root test to calculate the radius of convergence of the powerseries. You may freely use the fact that for all c > 0, n√c → 1 as n → ∞.[5 marks](c) Give an example of a power series with radius of convergence 53, justifyingyour answer.

(a) The ratio test involves taking the limit as n approaches infinity of the absolute value of the ratio of the (n+1)th term to the nth term of the series. However, in this case, the terms of the series do not have a consistent pattern due to the conditional definition of a_n. When n is a power of 2, a_n = 1, and when n is not a power of 2, a_n = 2. This means that the ratio of consecutive terms alternates between 1/2 and 2, and so the limit does not exist. Therefore, the ratio test is inconclusive and cannot be used to find the radius of convergence.

(b) The root test involves taking the limit as n approaches infinity of the nth root of the absolute value of the nth term of the series. In this case, the nth root of |a_n| is 1 when n is a power of 2 and is approximately 1.19 when n is not a power of 2. As n approaches infinity, the nth root of any constant c > 0 approaches 1. Therefore, the limit is 1, and by the root test, the radius of convergence is 1/1 = 1.

(c) An example of a power series with a radius of convergence of 5/3 is ∑(3/5)^n * x^n from n=0 to infinity. The ratio test gives the limit as n approaches infinity of |(3/5)^(n+1) * x^(n+1)| / |(3/5)^n * x^n| = |3/5| * |x|, which is less than 1 for |x| < 5/3. Therefore, the radius of convergence is 5/3.