Check if each of the following geometric series is convergent or divergent. Find the sum if it is convergent.

Question

a) 4+3+(9/4)+(27/16)+...

b) 2-(3/2)+(9/8)-(27/32)+...

c) -3+4-(16/3)+(64/9)+...

d) infinity: i=1 Σ (2^i)/(3^(i-1))

e) infinity: i=1 Σ (2^(2i))/(5^(i+1))

f) infinity: i=1 Σ (e^i)/(pi^(i-1))

Answer 1

(a)

Step 1:

The geometric series 

Common ratio .

term of the geometric series is .

Ratio test :

(i) If , then the series is absolutely convergent.

(ii) If or , then the series is divergent.

(iii) If , then the ratio test is inconclusive.

Here  and .

Find .

Hence by the ratio test, the series  is convergent.

Step 2:

The geometric series 

Find the sun of the infinite geometric series.

Sum of the infinite geometric series is .

Substitute and in the formula.

Therefore, sum of the series is 16.

Solution:

The series  is convergent.

Sum of the series is 16.

Answer 2

(b)

Step 1:

The geometric series .

Common ratio .

 term of the geometric series is .

Ratio test :

(i) If , then the series  is absolutely convergent.

(ii) If  or  , then the series  is divergent.

(iii) If , then the ratio test is inconclusive.

Here  and .

Find .

Hence by the ratio test, the series is convergent.

Step 2:

The geometric series .

Find the sum of the infinite geometric series.

Sum of the infinite geometric series is .

Substitute  and  in the formula.

Therefore, sum of the series is .

Solution:

The series is convergent.

Sum of the series is .

Answer 3

(c)

The geometric series .

Common ratio .

Thus .

Geometric series is divergent if .

Therefore, the series is divergent.

Answer 4

(d)

Step 1:

The geometric series  is .

The terms of the sequence are:

Substitute .

.

Substitute .

.

Substitute .

.

Therefore, the geometric sequence is  .

Common ratio .

Thus .

Geometric series is convergent if .

Therefore, the series is convergent.

Step 2:

The geometric series .

Find the sum of the infinite geometric series.

Sum of the infinite geometric series is .

Substitute and in the formula.

Therefore sum of the series is .

Solution:

Series is convergent.

Sum of the series is .

Answer 5

(e)

Step 1:

The geometric series  is .

The terms of the sequence are:

Substitute .

.

Substitute .

.

Substitute .

.

Therefore, the geometric sequence is  .

Common ratio .

Thus .

Geometric series is convergent if .

Therefore, the series is convergent.

Step 2:

The geometric series .

Find the sum of the infinite geometric series.

Sum of the infinite geometric series is .

Substitute and in the formula.

Therefore sum of the series is .

Solution:

Series is convergent.

Sum of the series is .

Answer 6

(f)

Step 1:

The geometric series  is .

The terms of the sequence are:

Substitute .

.

Substitute .

.

Substitute .

.

Therefore, the geometric sequence is  .

Common ratio .

Thus .

Geometric series is convergent if .

Therefore, the series is convergent.

Step 2:

The geometric series .

Find the sum of the infinite geometric series.

Sum of the infinite geometric series is .

Substitute and in the formula.

Therefore sum of the series is .

Solution:

Series is convergent.

Sum of the series is .