# Values of x for which a geometric series converges

• ellaingeborg
In summary, the conversation discusses a homework question involving a geometric sequence and series. The first three terms of the sequence are given, and the common ratio is found to be 2cos(x). The question asks to find the values of x for which the geometric series converges. The person seeking help is unsure of how to approach the problem, but has attempted to use a formula to find the sum of the series. It is mentioned that the geometric series converges if |r| < 1, and the person is reminded not to delete parts of the homework template in the future.
ellaingeborg
Need help with a homework question!
The question gives: The first three terms of a geometric sequence are sin(x), sin(2x) and 4sin(x)cos^2(x) for -π/2 < x < π/2.
First I had to find the common ratio which is 2cos(x)
Then the question asks to find the values of x for which the geometric series sinx + sin2x + ... converges.
I tried using
S∞= u/(1-r) where r is the common ratio, u is the first term of the sequence and S∞ the sum of the series.
this sum would equal
sinx/(1-2cosx)

ellaingeborg said:
Need help with a homework question!
The question gives: The first three terms of a geometric sequence are sin(x), sin(2x) and 4sin(x)cos^2(x) for -π/2 < x < π/2.
First I had to find the common ratio which is 2cos(x)
Then the question asks to find the values of x for which the geometric series sinx + sin2x + ... converges.
I tried using
S∞= u/(1-r) where r is the common ratio, u is the first term of the sequence and S∞ the sum of the series.
this sum would equal
sinx/(1-2cosx)

A geometric series converges if |r| < 1. What does this mean in relation to your geometric series?

In the future, please do not delete the three parts of the homework template. It is required for homework questions.

## What is a geometric series?

A geometric series is a series of numbers where each term is multiplied by a fixed number, called the common ratio, to get the next term. For example, 2, 4, 8, 16, 32, ... is a geometric series with a common ratio of 2.

## What is the formula for finding the sum of a geometric series?

The formula for finding the sum of a geometric series is S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio. This formula only works if the absolute value of the common ratio is less than 1, which is a necessary condition for the series to converge.

## What values of x make a geometric series converge?

A geometric series will converge if the absolute value of the common ratio is less than 1. This means that any value of x that makes the common ratio less than 1 will cause the series to converge.

## How do you test for convergence of a geometric series?

To test for convergence of a geometric series, you can use the ratio test or the root test. The ratio test states that if the absolute value of the common ratio is less than 1, the series will converge. The root test states that if the limit of the nth root of the terms in the series is less than 1, the series will converge.

## Can a geometric series diverge?

Yes, a geometric series can diverge if the absolute value of the common ratio is greater than or equal to 1. In this case, the terms in the series will continue to grow larger and larger, causing the series to diverge.

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