Consider the power series \(\sum\limits_{k = 1}^\infty {k{{\left( {\frac{x}{2}} \right)}^k}} \).

(i)     Find the radius of convergence.

(ii)     Find the interval of convergence.

Consider the infinite series \(\sum\limits_{k = 1}^\infty  {{{( - 1)}^{k + 1}} \times \frac{k}{{2{k^2} + 1}}} \).

(i)     Show that the series is convergent.

(ii)     Show that the sum to infinity of the series is less than 0.25.

(i)     consider \(\frac{{{T_{n + 1}}}}{{{T_n}}} = \frac{{\left| {\frac{{(n - 1){x^{n + 1}}}}{{{2^{n + 1}}}}} \right|}}{{\left| {\frac{{n{x^n}}}{{{2^n}}}} \right|}}\)     M1

\( = \frac{{(n + 1)\left| x \right|}}{{2n}}\)     A1

\( \to \frac{{\left| x \right|}}{2}{\text{ as }}n \to \infty \)     A1

the radius of convergence satisfies

\(\frac{R}{2} = 1\), i.e. R = 2     A1

(ii)     the series converges for \( - 2 < x < 2\), we need to consider \(x = \pm 2\)     (R1)

when x = 2, the series is \(1 + 2 + 3 + \ldots \)     A1

this is divergent for any one of several reasons e.g. finding an expression for or a comparison test with the harmonic series or noting that \(\mathop {\lim }\limits_{n \to \infty } {u_n} \ne 0\) etc.     R1

when x = – 2, the series is \( - 1 + 2 - 3 + 4 \ldots \)     A1

this is divergent for any one of several reasons

e.g. partial sums are

\( - 1,{\text{ }}1,{\text{ }} - 2,{\text{ }}2,{\text{ }} - 3,{\text{ }}3 \ldots \) or noting that \(\mathop {\lim }\limits_{n \to \infty } {u_n} \ne 0\) etc.     R1

the interval of convergence is \( - 2 < x < 2\)     A1

[10 marks]

(i)     this alternating series is convergent because the moduli of successive terms are monotonic decreasing     R1

and the \({n^{{\text{th}}}}\) term tends to zero as \(n \to \infty \)     R1

(ii)     consider the partial sums

0.333, 0.111, 0.269, 0.148, 0.246     M1A1

since the sum to infinity lies between any pair of successive partial sums, it follows that the sum to infinity lies between 0.148 and 0.246 so that it is less than 0.25     R1

Note: Accept a solution which looks only at 0.333, 0.269, 0.246 and states that these are successive upper bounds.

[5 marks]

Most candidates found the radius of convergence correctly but examining the situation when \(x = \pm 2\) often ended in loss of marks through inadequate explanations. In (b)(i) many candidates were able to justify the convergence of the given series. In (b)(ii), however, many candidates seemed unaware of the fact the sum to infinity lies between any pair of successive partial sums.

Most candidates found the radius of convergence correctly but examining the situation when \(x = \pm 2\) often ended in loss of marks through inadequate explanations. In (b)(i) many candidates were able to justify the convergence of the given series. In (b)(ii), however, many candidates seemed unaware of the fact the sum to infinity lies between any pair of successive partial sums.