Question

A series is given. (a) Find a formula for \(S_{n},\) the \(n^{\text {th }}\) partial sum of the series. (b) Determine whether the series converges or diverges. If it converges, state what it converges to. $$1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\frac{1}{81}+\cdots$$

Short answer

The series converges to \(\frac{3}{4}\).

Step-by-step solution

Identify the type of series

The given series is \(1 - \frac{1}{3} + \frac{1}{9} - \frac{1}{27} + \cdots\). We observe that it's a geometric series with the first term \(a = 1\) and common ratio \(r = -\frac{1}{3}\).

Write the formula for the nth-term

The general form for the nth-term of a geometric series is \(a_n = a \cdot r^{n-1}\). Therefore, in this series, each term is given by \((-1)^{n-1} \left( \frac{1}{3} \right)^{n-1}\).

Find the formula for the nth partial sum \(S_n\)

The formula for the sum of the first \(n\) terms of a geometric series is \(S_n = a \frac{1 - r^n}{1 - r}\) for \(r eq 1\). Substituting \(a = 1\) and \(r = -\frac{1}{3}\), we get \[ S_n = \frac{1 - \left(-\frac{1}{3}\right)^n}{1 - \left(-\frac{1}{3}\right)} = \frac{1 - \left(-\frac{1}{3}\right)^n}{\frac{4}{3}} = \frac{3}{4} \left(1 - \left(-\frac{1}{3}\right)^n\right). \]

Check for convergence

A geometric series converges if \(|r| < 1\). Here, \(r = -\frac{1}{3}\) and \(|r| = \frac{1}{3} < 1\), so the series converges.

Determine the sum of the infinite series

For a convergent geometric series, the sum to infinity is given by \(S = \frac{a}{1 - r}\). Substituting the values for \(a = 1\) and \(r = -\frac{1}{3}\), we find \[ S = \frac{1}{1 - (-\frac{1}{3})} = \frac{1}{1 + \frac{1}{3}} = \frac{1}{\frac{4}{3}} = \frac{3}{4}. \]

Key concepts

Partial Sum

Understanding the concept of a partial sum is crucial when dealing with series. A partial sum, denoted as \(S_n\), represents the sum of the first \(n\) terms of a series. In the case of a geometric series like the one given, each term follows a specific pattern. This allows us to use a formula to find the partial sum, making calculations much simpler.

• The formula for the \(n\)-th partial sum \(S_n\) of a geometric series is: \(S_n = a \frac{1 - r^n}{1 - r}\)
• Here, \(a\) is the first term and \(r\) is the common ratio.

For the series \(1 - \frac{1}{3} + \frac{1}{9} - \frac{1}{27} + \cdots\), we have \(a = 1\) and \(r = -\frac{1}{3}\). By substituting these values into the formula, we obtain the partial sum:\[ S_n = \frac{3}{4} \left(1 - \left(-\frac{1}{3}\right)^n\right). \] This expression helps us easily calculate the sum of any number of terms in the series.

Convergence

In mathematics, convergence refers to the behavior of a series as the number of terms increases. A series is convergent if the sum of its terms approaches a finite number. Understanding whether a series converges is vital for determining if it has a meaningful sum.

• A geometric series converges if the absolute value of its common ratio \(|r|\) is less than 1.
• Conversely, if \(|r|\geq 1\), the series diverges.

For the given series \(1 - \frac{1}{3} + \frac{1}{9} - \frac{1}{27} + \cdots\), the common ratio is \(r = -\frac{1}{3}\). Since \(|r| = \frac{1}{3} < 1\), the series is convergent. This means that as we sum more and more terms, the series tends to settle towards a particular value, rather than growing indefinitely.

Geometric Series Formula

The geometric series formula is a powerful tool that lets us quickly calculate the sum of terms in a geometric series. It applies to series where each term is a constant multiple of the previous one.

• The general formula for the nth-term of a geometric series is \(a_n = a \cdot r^{n-1}\).
• For the sum of the first \(n\) terms, the formula is: \(S_n = a \frac{1 - r^n}{1 - r}\).

In our specific series, the first term \(a\) is 1, and the common ratio \(r\) is \(-\frac{1}{3}\). Substituting these into the partial sum formula gives us the equation for finding the sum of the first \(n\) terms: \[ S_n = \frac{3}{4} \left(1 - \left(-\frac{1}{3}\right)^n\right) \]This standardized approach greatly simplifies the process of working with geometric series, making them less daunting to analyze and manipulate.

Infinite Series Sum

The concept of an infinite series sum is where mathematics gets particularly interesting. When a series is convergent, it is possible to determine the sum of an infinite number of terms.

• If a geometric series converges, the sum to infinity can be determined using the formula: \(S = \frac{a}{1 - r}\).
• This formula is only valid if \(|r| < 1\), indicating a convergent series.

In the case of our given series, with \(a = 1\) and \(r = -\frac{1}{3}\), we apply the formula and find:\[ S = \frac{1}{1 - (-\frac{1}{3})} = \frac{1}{\frac{4}{3}} = \frac{3}{4} \]This result tells us the entire sum of the series, extending infinitely, approaches \(\frac{3}{4}\). Understanding the infinite series sum provides deep insights into the behavior of convergent series.