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 sumof the series is 16.

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 sumof the series is .

(c)

Step 1:

The geometric series .

Common ratio .

Thus .

Geometric series is divergent if .

Therefore, the series is divergent.

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 sumof the series is .

(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 .

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 sumof the series is .

(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 .

(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 .

.