Consider the infinite series

\[\frac{1}{{2\ln 2}} - \frac{1}{{3\ln 3}} + \frac{1}{{4\ln 4}} - \frac{1}{{5\ln 5}} +  \ldots {\text{ .}}\]

(a)     Show that the series converges.

(b)     Determine if the series converges absolutely or conditionally.

(a)     applying the alternating series test as \(\forall n \geqslant 2,\frac{1}{{n\ln n}} \in {\mathbb{R}^ + }\)     M1

\(\forall n,\frac{1}{{(n + 1)\ln (n + 1)}} \leqslant \frac{1}{{n\ln n}}\)     A1

\(\mathop {\lim }\limits_{n \to \infty } \frac{1}{{n\ln n}} = 0\)     A1

hence, by the alternating series test, the series converges     R1

[4 marks]

(b)     as \(\frac{1}{{x\ln x}}\) is a continuous decreasing function, apply the integral test to determine if it converges absolutely     (M1)

\(\int_2^\infty  {\frac{1}{{x\ln x}}{\text{d}}x = \mathop {\lim }\limits_{b \to \infty } \int_2^b {\frac{1}{{x\ln x}}{\text{d}}x} } \)     M1A1

let \(u = \ln x\) then \({\text{d}}u = \frac{1}{x}{\text{d}}x\)     (M1)A1

\(\int {\frac{1}{u}{\text{d}}u = \ln u} \)     (A1)

hence, \(\mathop {\lim }\limits_{b \to \infty } \int_2^b {\frac{1}{{x\ln x}}{\text{d}}x = \mathop {\lim }\limits_{b \to \infty } \left[ {\ln (\ln x)} \right]_2^b} \) which does not exist     M1A1A1

hence, the series does not converge absolutely     (A1)

the series converges conditionally     A1

[11 marks]

Total [15 marks]

Part (a) was answered well by many candidates who attempted this question. In part (b), those who applied the integral test were mainly successful, but too many failed to supply the justification for its use, and proper conclusions.