For which values of x does the power series converge?

Which of the following series is convergent

(a) Find the two possible values of x  .For each of your values of x ,(b) find the corresponding value of the common ratio of the series.Given that the series is convergent,(c ) find the sum to infinity of the series.

Express √1+2𝑥√1−3𝑥3 as a power series as far as the term 𝑥2, state the range of values of 𝑥 forwhich the series is convergent

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 geometric series has first term (11x−3), Second term (5x+3) and third term (3x−3) . (a) Find the two possible values of x  .For each of your values of x ,(b) find the corresponding value of the common ratio of the series.Given that the series is convergent,(c ) find the sum to infinity of the series.Please give 1 answer.(a) x=0   or  x= 9(b) x=0 ,r=−1      x=9 ,   r=12 (c) 192( a ) x=3    or x=8 (b) x=3 , r=−2      x=8 , r=23 (c ) 148(a)  x=1   or x=−9 (b)x=1 , r=1     x=−9 , r=54 (c) 129(a) x=2 or x=−7 (b) x=2 , r=23     x=−7, r=5 (c) 196