Answer
Fill the blaks with
a. ... $a\displaystyle \cdot\frac{1-r^{n}}{1-r}$ ...
b. ... geometric... converges ...$\displaystyle \frac{a}{1-r}$... diverges
Work Step by Step
a. See p. 861.
For the geometric sequence $a_{n}=ar^{n-1}$
the nth partial sum$ S_{n}=\displaystyle \sum_{k=1}^{n}ar^{k-1}$ (where $r\neq 1$)
is given by$ \displaystyle \quad S_{n}=a\cdot\frac{1-r^{n}}{1-r}$
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Fill the blank with $a\displaystyle \cdot\frac{1-r^{n}}{1-r}$
b.
See p. 863.
An infinite geometric series is a series of the form
$ a+ar+ar^{2}+ar^{3}+\cdots+ar^{n-1}+\cdots$
An infinite geometric series for which $|r| < 1$
converges, and has the sum $S=\displaystyle \frac{a}{1-r}$
If $|r| \geq 1$, the series diverges (the sum does not exist).
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Fill the blanks with
... geometric... converges ...$\displaystyle \frac{a}{1-r}$... diverges
