Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.
-$$\frac { 7 } { 1 \cdot 2 } + \frac { 7 } { 2 \cdot 3 } + \frac { 7 } { 3 \cdot 4 } + \ldots + \frac { 7 } { \mathrm { n } ( \mathrm { n } + 1 ) } + \ldots$$
A) $\frac { 7 ( \mathrm { n } + 1 ) } { \mathrm { n } } ; 7$
B) $\frac { 7 ( n + 2 ) } { n + 1 } ; 7$
C) $\frac { 7 ( n + 1 ) } { n + 2 } ; 7$
D) $\frac { 7 \mathrm { n } } { \mathrm { n } + 1 } ; 7$

Question

Quizplus · Accepted Answer

The given series can be rewritten using partial fraction decomposition as $\frac{7}{n(n+1)} = 7\left(\frac{1}{n} - \frac{1}{n+1}\right)$. Summing these terms from 1 to n results in a telescoping series, where all terms cancel out except the first and the last, leading to a sum of $\frac{7n}{n+1}$. As $n$ approaches infinity, the sum converges to 7.

Find a Formula for the Nth Partial Sum of the Series