Answer
Please see step-by-step
Work Step by Step
$\displaystyle \lim_{n\rightarrow\infty}a_{n}=5$ means that
the limit of the sequence $\{a_{n}\}$ is 5.
(As n gets very large, $a_{n}$ approaches the value 5)
$\displaystyle \sum_{n=1}^{\infty}a_{n}=5$
(see definitions of Convergent and Divergent Series (page 595))
For the infinite series $\displaystyle \sum_{n=1}^{\infty}a_{n}$, the $n$ th partial sum is $ S_{n}=a_{1}+a_{2}+\cdot \cdot +a_{n}$.
$\displaystyle \sum_{n=1}^{\infty}a_{n}=5$ means that
the limit of the sequence of partial sums $\{S_{n}\}$ is 5.
