Given that $\mathrm{f}\left( r \right)=\left( r+2 \right){{2}^{-r}}$, where $r\in \mathbb{Z}$, show

Question

Given that $\mathrm{f}\left( r \right)=\left( r+2 \right){{2}^{-r}}$, where $r\in \mathbb{Z}$, show that $\mathrm{f}\left( r \right)-\mathrm{f}\left( r-2 \right)={{2}^{-r}}\left( 2-3r \right)$. [1] Show that $\sum\limits_{r=1}^{n}{{{2}^{-r}}\left( 2-3r \right)}={{2}^{-n}}\left( 3n+4 \right)-4$. [3] Hence find $\sum\limits_{r=1}^{n}{r{{2}^{-r}}}$. [4] Determine, with a reason, if the series $\sum\limits_{r=1}^{\infty }{r{{2}^{-r}}}$converges. [2]

Accepted Answer

(ii) $\sum_{r=1}^{n} r \cdot 2^{-r} = 2 - 2^{-n}(n+2)$ (iii) Series converges to $2$

2018 TJC Promo Q7

2018 TJC Promo Q7

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