Each term of the power series \(\frac{1}{{1 \times 2}} + \frac{1}{{4 \times 5}}x + \frac{1}{{7 \times 8}}{x^2} + \frac{1}{{10 \times 11}}{x^3} +  \ldots \) has the form \(\frac{1}{{b(n) \times c(n)}}{x^n}\), where \(b(n)\) and \(c(n)\) are linear functions of \(n\).

(a)     Find the functions \(b(n)\) and \(c(n)\).

(b)     Find the radius of convergence.

(c)     Find the interval of convergence.

(a)     \(b(n) = 3n + 1\)     A1

\(c(n) = 3n + 2\)     A1

Note: \(b(n)\) and \(c(n)\) may be reversed.

[2 marks]

(b)     consider the ratio of the \({(n + 1)^{{\text{th}}}}\) and \({n^{{\text{th}}}}\) terms:     M1

\(\frac{{3n + 1}}{{3n + 4}} \times \frac{{3n + 2}}{{3n + 5}} \times \frac{{{x^{n + 1}}}}{{{x^n}}}\)     A1

\(\mathop {{\text{lim}}}\limits_{n \to 0}  \frac{{3n + 1}}{{3n + 4}} \times \frac{{3n + 2}}{{3n + 5}} \times \frac{{{x^{n + 1}}}}{{{x^n}}}x\)     A1

radius of convergence: \(R = 1\)     A1

[4 marks]

(c)     any attempt to study the series for \(x = -1\) or \(x = 1\)     (M1)

converges for \(x = 1\) by comparing with p-series \(\sum {\frac{1}{{{n^2}}}} \)     R1

attempt to use the alternating series test for \(x = -1\)     (M1)

Note: At least one of the conditions below needs to be attempted for M1.

\(\left| {{\text{terms}}} \right| \approx \frac{1}{{9{n^2}}} \to 0\) and terms decrease monotonically in absolute value     A1

series converges for \(x = -1\)     R1

interval of convergence: \(\left[ { - 1,{\text{ 1}}} \right]\)     A1

Note: Award the R1s only if an attempt to corresponding correct test is made;

     award the final A1 only if at least one of the R1s is awarded;

     Accept study of absolute convergence at end points.

[6 marks]