Understanding Infinite Geometric Series

What is the main difference between a geometric sequence and a geometric series?

• A sequence is always increasing, while a series is always decreasing.

• A sequence involves addition, while a series involves multiplication.

• A sequence is always finite, while a series is always infinite.

• A sequence is a list of numbers, while a series is the sum of those numbers.

Which of the following conditions will cause a geometric series to diverge?

• The common ratio is a fraction.

• The common ratio is greater than or equal to 1 or less than or equal to -1.

• The common ratio is exactly 0.

• The common ratio is between -1 and 1.

What is the formula for the sum of an infinite geometric series?

• Sum = n * a, where n is the number of terms and a is the first term.

• Sum = a / (1 - r), where a is the first term and r is the common ratio.

• Sum = a * r^n, where a is the first term and r is the common ratio.

• Sum = a + r, where a is the first term and r is the common ratio.

In the example provided, what is the sum of the infinite geometric series with the first term 5 and common ratio 1/2?

• 15

• 10

• 20

• 5

Why does the formula for the sum of an infinite geometric series only work when the common ratio is between -1 and 1?

• Because it makes the series diverge.

• Because it ensures the series is always increasing.

• Because the formula is undefined for other values.

• Because only then does the series have a finite sum.