Q. 2 - Calculus

Question

Create illustrations of the thing(s) mentioned in the following. Look for examples that are distinct from those in the text.

(a) A geometric sequence that converges  With $r <0$, $\sum _{k=0}^{\infty }c{r}^k$  .

(b) A Divergent geometric sequences  With $r< 0$,$\sum _{k=0}^{\infty }c{r}^k$

(c) An unrelated divergent series to a geometric series

Step-by-Step Solution

Answer

1. The convergent geometric series is $\sum _{k=0}^{\infty }c{r}^k$with $r<0$ is $\sum _{k=0}^{\infty }{\left(-\frac{10}{11}\right)}^k.$
2. The divergent geometric series is $\sum _{k=0}^{\infty }c{r}^k$with $r<0$ is $\sum _{k=0}^{\infty }(-10{)}^k.$
3. A diverging series, that is not geometric,  is $\sum _{k=0}^{\infty }\left(k^2+1\right).$

Part (a) Step 1: Given information.

A convergent geometric series is $\sum _{k=0}^{\infty }c{r}^k$with $r<0$

Part(a) Step 2: Calculation

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

The series is $\sum _{k=0}^{\infty }{a}_k=\sum _{k=0}^{\infty }{\left(-\frac{10}{11}\right)}^k.$

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 $\sum _{k=0}^{\infty }c{r}^k$with $r<0$ is

$\sum _{k=0}^{\infty }{\left(-\frac{10}{11}\right)}^k.$

Part(b) Step 1: Given information

A divergent geometric series is $\sum _{k=0}^{\infty }c{r}^k$ with $r<0$

Part(b) Step 2: Calculation

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

The series is  $\sum _{k=0}^{\infty }{a}_k=\sum _{k=0}^{\infty }(-10{)}^k$.

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 $\sum _{k=0}^{\infty }c{r}^k$ with $r<0$is $\sum _{k=0}^{\infty }(-10{)}^k.$

Part(c) Step 1: Given information

A non-geometric divergent series

Part(c) Step 2: Calculation

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

The series is $\sum _{k=0}^{\infty }{a}_k=\sum _{k=0}^{\infty }\left(k^2+1\right).$

Though divergent, the sequence is not geometric.

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

$\sum _{k=0}^{\infty }\left(k^2+1\right).$