Question

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ 1-\frac{3}{4}+\frac{9}{16}-\frac{27}{64}+\cdots $$

Short answer

The series converges, and its sum is \( \frac{4}{7} \).

Step-by-step solution

Identify the first term

The first term of the series is given directly. Here, the first term (denoted as \(a\)) is \(1\).

Find the common ratio

The common ratio (denoted as \(r\)) is found by dividing the second term by the first term: \(-\frac{3}{4} \div 1 = -\frac{3}{4}\). Then verify the ratio by checking successive terms: \(\frac{9}{16} \div -\frac{3}{4} = -\frac{3}{4}\) and so on. Common ratio is consistent.

Determine convergence or divergence

To determine if an infinite geometric series converges, the absolute value of the common ratio must be less than 1, i.e., \(|r| < 1\). Here, \(|-\frac{3}{4}| = \frac{3}{4} < 1\), so the series converges.

Find the sum of the series

For a convergent geometric series, the sum \(S\) is given by the formula: \[ S = \frac{a}{1 - r} \]. Substituting \(a = 1\) and \(r = -\frac{3}{4}\) into the formula, \[ S = \frac{1}{1 - (-\frac{3}{4})} = \frac{1}{1 + \frac{3}{4}} = \frac{1}{\frac{7}{4}} = \frac{4}{7} \].

Key concepts

Convergence and Divergence

An infinite geometric series can either converge or diverge. It all depends on the value of its **common ratio**. To determine this, you need to examine the absolute value of the common ratio.

• If \(|r| < 1\), the series converges. This means that the sum of the series approaches a finite value as the number of terms increases.
• If \(|r| \geq 1\), the series diverges. This means the series doesn't settle towards a finite value, often growing without bound.

This concept is crucial for understanding whether it's possible to find a sum for the series.In our example, the common ratio is \(-\frac{3}{4}\). Because \(|-\frac{3}{4}| = \frac{3}{4} < 1\), the series converges.If you encounter other series, always first find out whether \(|r| < 1\) before calculating the sum.

Common Ratio

The common ratio is a key factor in geometric series, determining how we move from one term to the next. To find the common ratio, simply divide a term in the series by the previous term.For example, in the series given, \[-\frac{3}{4} \div 1 = -\frac{3}{4},\] then double-check it by using other consecutive terms:\[\frac{9}{16} \div -\frac{3}{4} = -\frac{3}{4}.\]It's very important to verify the common ratio with multiple terms to ensure consistency.Understanding the common ratio helps identify whether the series grows or shrinks as you go through the terms. This directly influences whether the series converges or diverges.

Sum of Series

When a geometric series converges, we can calculate its sum using a simple formula.The sum \(S\) of an infinite convergent geometric series is given by:\[ S = \frac{a}{1 - r}, \] where \(a\) is the first term and \(r\) is the common ratio.For the series provided:\[- S = \frac{1}{1 - (-\frac{3}{4})} = \frac{1}{1 + \frac{3}{4}} = \frac{1}{\frac{7}{4}} = \frac{4}{7}.\]This formula transforms the seemingly endless sum into a very manageable number.Therefore, for the series given, the sum is \frac{4}{7}\. This showcases how powerful the formula is and why finding the common ratio and checking for convergence first is so important.