Find $\sum\limits_{r=1}^{k}{\left[ {{\left( -\frac{1}{2} \right)}^{r+1}}+\ln \left( r+1 \right) \rig

Question

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] Hence determine if $\sum\limits_{r=1}^{\infty }{\left[ {{\left( -\frac{1}{2} \right)}^{r+1}}+\ln \left( r+1 \right) \right]}$ exists. [1] 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 ${{13}^{\mathrm{th}}}$ term of the series. [4]

Accepted Answer

(a)(i) $\frac16\left[1-\left(-\frac12\right)^k\right]+\ln\left((k+1)!\right)$ (a)(ii) The sum to infinity does not exist. (b) $T_{13}=-13.5$

2022 HCI P2 Q4

2022 HCI P2 Q4

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