Write down each of the following series in summation notation $0!+1!+2!+3!+…+6!$ $1!+3!+5!+7!…$

Question

Write down each of the following series in summation notation $0!+1!+2!+3!+…+6!$ $1!+3!+5!+7!…$ to $n$ terms $1\cdot 2\cdot 3+2\cdot 3\cdot 4+3\cdot 4\cdot 5+4\cdot 5\cdot 6+…$ to $n$ terms $-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}+…$ to $n+1$ terms $2\cdot 3\cdot 4-3\cdot 4\cdot 5+4\cdot 5\cdot 6-5\cdot 6\cdot 7+…$ to $n$ terms $-\frac{3}{2}+\frac{5}{4}-\frac{7}{8}+\frac{9}{16}+…$ to $n+1$ terms $2+6+18+54+…$ to $n$ terms $-\frac{1}{2}+3-\frac{1}{4}+5-\frac{1}{6}+7-\frac{1}{8}…$ to $n$ terms

Accepted Answer

(i) $\sum_{r=0}^{6}r!$ (ii) $\sum_{r=1}^{n}(2r-1)!$ (iii) $\sum_{r=1}^{n}r(r+1)(r+2)$ (iv) $\sum_{r=1}^{n+1}\left(-\frac{1}{2}\right)^r$ (v) $\sum_{r=1}^{n}(-1)^{r+1}(r+1)(r+2)(r+3)$ (vi) $\sum_{r=1}^{n+1}(-1)^r\frac{2r+1}{2^r}$ (vii) $\sum_{r=1}^{n}2\cdot3^{r-1}$ (viii) $\sum_{r=1}^{n}(-1)^r(r+1)^{(-1)^r}$

Writing the series in sigma notation

Writing the series in sigma notation

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