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.

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.