FOR WHAT VALUE OF X DOES THE GEOMETRIC SERIES CONVERGE

Each of the following represents an infinite geometric series. For what values of x will each series be convergent?

Problem 1 : 

5 + 5x + 5x² + 5x³ + …

Solution :

5 + 5x + 5x² + 5x³ + …

a₁ = 5

r = a₂ / a₁

r = 5x / 5

r = x

This is a geometric series with common ratio x. The value of r = x is in the interval -1 < x < 1.

Problem 2 :

1 + x/3 + x²/9 + x³/27 + …

Solution :

1 + x/3 + x²/9 + x³/27 + …

a₁ = 1

r = a₂ / a₁

r = (x/3) / 1

r = x/3

This is a geometric series with common ratio x/3, and the converge condition is

|x/3| < 1

-1 < x/3 < 1

-3 < x < 3

So, when -3 < x < 3, the series converges.

Problem 3 :

2 + 4x + 8x² + 16x³ + …

Solution :

2 + 4x + 8x² + 16x³ + …

a₁ = 2

r = a₂ / a₁

r = 4x / 2

r = 2x

This is a geometric series with common ratio 2x, and the converge condition is

|2x| < 1

-1 < 2x < 1

-1/2 < x < 1/2

Problem 4 :

The infinite series given by

1 + 3x + 9x² + 27x³ + …

has a sum of 4. What is the value of x?

List the first four terms of the series.

Solution :

1 + 3x + 9x² + 27x³ + …

a₁ = 1

r = a₂ / a₁

r = 3x / 1

r = 3x

This is a geometric series with common ratio 3x.

S_∞ = a₁ / (1 - r)

4 = 1 / (1 - 3x)

4(1 - 3x) = 1

4 - 12x = 1

-12x = -3

x = 1/4

The first four terms are

= 1 + 3(1/4) + 9(1/4)² + 27(1/4)³ + …

= 1 + 3/4 + 9/16 + 27/64 + …

Problem 5 :

Dominique and Rita are discussing the series

-1/3 + 4/9 - 16/27 + ...

Dominique says that the sum of the series is -1/7.

Rita says that the series is divergent and has no sum.

a. Who is correct?

b. Explain your reasoning.

Solution :

Checking whether it is convergent or divergent :

-1/3 + 4/9 - 16/27 +...

a₁ = -1/3

r = a₂ / a₁

r = (4/9) / (-1/3)

r = -4/3

The value of r = -4/3 is not in the interval -1 < r < 1.

So, the given series is divergent.

a)  Rita is correct, because the given series is divergent.

b)  Since it is divergent, we can not figure out the sum.