# Write the series and find the sum of the series of sigma notation. sum_{i=0}^6frac{i}{i-1}

Question
Series
Write the series and find the sum of the series of sigma notation.
$$\displaystyle{\sum_{{{i}={0}}}^{{6}}}{\frac{{{i}}}{{{i}-{1}}}}$$

2021-03-10
The given series is $$\displaystyle{\sum_{{{i}={0}}}^{{6}}}{\frac{{{i}}}{{{i}-{1}}}}$$
Expand the given series as follows.
$$\displaystyle{\sum_{{{i}={0}}}^{{6}}}{\frac{{{i}}}{{{i}-{1}}}}={\frac{{{0}}}{{{0}-{1}}}}+{\frac{{{1}}}{{{1}-{1}}}}+{\frac{{{2}}}{{{2}-{1}}}}+{\frac{{{3}}}{{{3}-{1}}}}+{\frac{{{4}}}{{{4}-{1}}}}+{\frac{{{5}}}{{{5}-{1}}}}+{\frac{{{6}}}{{{6}-{1}}}}$$
$$\displaystyle={0}+{\frac{{10}}{+}}{\frac{{21}}{+}}{\frac{{32}}{+}}{\frac{{43}}{+}}{\frac{{54}}{+}}{\frac{{65}}{}}$$
$$\displaystyle={\frac{{10}}{+}}{\frac{{21}}{+}}{\frac{{32}}{+}}{\frac{{43}}{+}}{\frac{{54}}{+}}{\frac{{65}}{}}$$
Thus, the expanded form of the given series is $$\displaystyle{\frac{{10}}{+}}{\frac{{21}}{+}}{\frac{{32}}{+}}{\frac{{43}}{+}}{\frac{{54}}{+}}{\frac{{65}}{}}$$
Simplify further and evaluate the sum of the series as follows.
$$\displaystyle{\sum_{{{i}={0}}}^{{6}}}={\frac{{10}}{+}}{\frac{{21}}{+}}{\frac{{32}}{+}}{\frac{{43}}{+}}{\frac{{54}}{+}}{\frac{{65}}{}}$$
$$\displaystyle=\text{undefined}+{\frac{{21}}{+}}{\frac{{32}}{+}}{\frac{{43}}{+}}{\frac{{54}}{+}}{\frac{{65}}{\quad}}{\left({\frac{{10}}{=}}\text{undefined or }\ \infty\right)}$$
=undefined
Thus, the sum of the given series is not defined.

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