2022 HCI P2 Q4
(a)
(i) Find $\sum\limits_{r=1}^{k}{\left[ {{\left( -\frac{1}{2} \right)}^{r+1}}+\ln \left( r+1 \right) \right]}$ in terms of $k$. Simplify your answer.
[4]
(ii) Hence determine if $\sum\limits_{r=1}^{\infty }{\left[ {{\left( -\frac{1}{2} \right)}^{r+1}}+\ln \left( r+1 \right) \right]}$ exists.
[1]
(b)
The first term of an arithmetic series is positive. The sum of the first $6$ terms of the series is $4.5$, and the product of the first four terms of the series is $0$. Find the ${{13}^{\text{th}}}$ term of the series.
[4]
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