91Ó°ÊÓ

A convergent geometric series is $\underset{k=0}{\overset{∞}{∑}}c{r}^k$with $r<0$

Finding the series that meets the specified criterion is the goal.

The series is $\underset{k=0}{\overset{∞}{∑}}{a}_k=\underset{k=0}{\overset{∞}{∑}}{\left(-\frac{10}{11}\right)}^k.$

The series is geometric series with ratio $r=-\frac{10}{11}$which is less than 0.

The geometric series is convergent because ratio $r=-\frac{10}{11}$is less than one.

The following is an illustration of a convergent geometric series $\underset{k=0}{\overset{∞}{∑}}c{r}^k$with $r<0$is

$\underset{k=0}{\overset{∞}{∑}}{\left(-\frac{10}{11}\right)}^k.$

A divergent geometric series is $\underset{k=0}{\overset{∞}{∑}}c{r}^k$ with $r<0$

Finding the series that meets the specified criterion is the goal.

The series is $\underset{k=0}{\overset{∞}{∑}}{a}_k=\underset{k=0}{\overset{∞}{∑}}(-10{)}^k$.

The series is geometric series with ratio $r=-10$which is less than 0.

The geometric series is divergent because ratio $r=-10$is less than 0.

The following is an illustration of a divergent geometric series $\underset{k=0}{\overset{∞}{∑}}c{r}^k$with $r<0$is $\underset{k=0}{\overset{∞}{∑}}(-10{)}^k.$

A non-geometric divergent series

Finding the series that meets the specified criterion is the goal.

The series is $\underset{k=0}{\overset{∞}{∑}}{a}_k=\underset{k=0}{\overset{∞}{∑}}\left(k^2+1\right).$

Though divergent, the sequence is not geometric.

Consequently, an illustration of a diverging series that is not geometric.

$\underset{k=0}{\overset{∞}{∑}}\left(k^2+1\right).$