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. $$\sum_{n=0}^{\infty} \frac{5}{2^{n}}$$

Short answer

The series converges with the nth partial sum \( S_n = 10 \left( 1-\left(\frac{1}{2}\right)^{n+1} \right) \) and the total sum is 10.

Step-by-step solution

Identify the Type of Series

The given series \( \sum_{n=0}^{\infty} \frac{5}{2^{n}} \) is a geometric series. A geometric series has the form \( \sum_{n=0}^{\infty} ar^n \), where \(a\) is the first term and \(r\) is the common ratio.

Find the Common Ratio

For the geometric series \( \sum_{n=0}^{\infty} \frac{5}{2^{n}} \), the first term is \( a = 5 \). The common ratio \( r \) can be found from the term \( \frac{5}{2^n} \), which is \( \frac{1}{2} \).

Write the Formula for the nth Partial Sum

The formula for the nth partial sum \( S_n \) of a geometric series \( \sum_{n=0}^{\infty} ar^n \) is \( S_n = a \frac{1-r^{n+1}}{1-r} \). Substituting the values \( a = 5 \) and \( r = \frac{1}{2} \), we have: \[ S_n = 5 \frac{1-(\frac{1}{2})^{n+1}}{1-\frac{1}{2}} = 5 \times 2 \left( 1-\left(\frac{1}{2}\right)^{n+1} \right) = 10 \left( 1-\left(\frac{1}{2}\right)^{n+1} \right) \]

Determine Convergence of the Series

A geometric series converges if the absolute value of the common ratio \( |r| < 1 \). In this case, \( r = \frac{1}{2} \) and \( |\frac{1}{2}| < 1 \), so the series converges.

Find the Sum of the Convergent Series

For a convergent geometric series, the sum \( S \) is given by \( S = \frac{a}{1-r} \). Since \( a = 5 \) and \( r = \frac{1}{2} \), \[ S = \frac{5}{1 - \frac{1}{2}} = 5 \times 2 = 10 \]. Therefore, the series converges to 10.

Key concepts

Convergence of Series

When we talk about the convergence of a series, we are asking whether the sum of the infinite series approaches a finite limit.
For a geometric series, which has the form \( \sum_{n=0}^{\infty} ar^n \), the series converges if the absolute value of the common ratio \( |r| \) is less than 1.
This condition ensures that each subsequent term becomes progressively smaller, causing the overall sum to stabilize towards a certain value. In the original exercise, we have the series \( \sum_{n=0}^{\infty} \frac{5}{2^n} \). Here, we identified it as a geometric series, where the common ratio \( r = \frac{1}{2} \) satisfies the convergence condition, \( |r| < 1 \).
Since \( |\frac{1}{2}| = 0.5 \) is indeed less than 1, this series converges.

Partial Sum Formula

The formula for calculating the partial sum \( S_n \) of a geometric series is an essential tool in determining whether a series converges and finding its sum.
For a geometric series \( \sum_{n=0}^{\infty} ar^n \), the \( n^{\text{th}} \) partial sum is given by:

• \( S_n = a \frac{1-r^{n+1}}{1-r} \)

This formula allows us to calculate the sum of the first \( n \) terms of the series by plugging in the values for \( a \) (the first term) and \( r \) (the common ratio).
In the original exercise, using the formula for the partial sum where \( a = 5 \) and \( r = \frac{1}{2} \), we found:

• \( S_n = 5 \cdot 2 \left(1 - \left(\frac{1}{2}\right)^{n+1}\right) \)
• \( S_n = 10 \left(1 - \left(\frac{1}{2}\right)^{n+1}\right) \)

This equation gives us a detailed snapshot of how the series' partial sums behave as \( n \) increases.

Geometric Progression

A geometric progression is a sequence where each term after the first is found by multiplying the previous term by a constant, known as the common ratio \( r \).
It is the backbone of a geometric series. In mathematical terms, if \( a \) is the first term, the sequence is represented as:

• \( a, ar, ar^2, ar^3, \ldots \)

As each step multiplies the previous term by \( r \), this characteristic ratio drives the progression.In the context of the original exercise, the geometric series \( \sum_{n=0}^{\infty} \frac{5}{2^n} \) has its terms in geometric progression, starting from the first term \( a = 5 \).
The common ratio here is \( r = \frac{1}{2} \), which highlights the consistency in scaling from one term to the next.
Understanding geometric progression is crucial because it allows us to easily determine the nature of the series—whether it is convergent or divergent—and compute various sums associated with the series effectively.