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

For which values of x does the power series converge?

Determine the radius of convergence of the power series∞Xn=0−√2nx2n

Which of the following series is convergen

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.

Determine if the following series converges or diverges. If it converges determine its sum.∞∑n=2 1n2−1