Question

Evaluate. $$ \sum_{n=1}^{\infty} \frac{2^{n+4}}{7^{n}} $$

Short answer

The sum is 22.4.

Step-by-step solution

Identify the Series

The given series is \( \sum_{n=1}^{ ext{∞}} \frac{2^{n+4}}{7^{n}} \). This can be rewritten to identify the pattern of a geometric series. We notice that the series can be broken down into factors to match the form of a geometric series in the standard form \( \sum_{n=1}^{ ext{∞}} ar^n \).

Factor Out Constant Terms

Notice that \( 2^{n+4} = 2^4 \times 2^n \). Therefore, rewrite the series as \( \sum_{n=1}^{ ext{∞}} \frac{16 \times 2^n}{7^n} = 16 \sum_{n=1}^{ ext{∞}} (\frac{2}{7})^n \). Now the series is in the standard form \( ar^n \) with \( a = 1 \) and \( r = \frac{2}{7} \).

Apply Formula for Infinite Geometric Series

The formula for the sum \( S \) of an infinite geometric series \( ar^n \) is \( \frac{a}{1-r} \), provided \( |r| < 1 \). In this case, \( a = 1 \) and \( r = \frac{2}{7} \). So the sum is \( \frac{1}{1 - \frac{2}{7}} = \frac{1}{\frac{5}{7}} = \frac{7}{5} \).

Calculate the Final Sum

Multiply the sum of the geometric series by the factored constant (16) from the earlier step. Thus, the sum of the original series is \( 16 \times \frac{7}{5} = \frac{112}{5} = 22.4 \).

Key concepts

Infinite Series

An infinite series is the sum of the terms of an infinite sequence. Unlike in finite sequences, where the number of terms is limited, in an infinite series, terms continue indefinitely. This implies that the series is constructed by infinitely adding successive terms:

• The representation of infinite series often looks like \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) denotes the sequence's terms.
• An infinite series can have a finite sum depending on whether it converges.

Understanding infinite series is crucial for various mathematical concepts, including calculus and analysis, where infinite processes are commonplace. The primary focus is often on whether and how an infinite series converges to a particular value.

Sum of Series

Calculating the sum of an infinite series might seem daunting because of its never-ending nature. However, specific methods allow us to find the sum when the series is geometric:

• A geometric series is a series where each term is a constant multiple of the previous term. It's generally written as \( \sum_{n=0}^{\infty} ar^n \), with \( a \) being the first term and \( r \) the common ratio.
• For a geometric series, the sum \( S \) can be calculated using the formula \( \frac{a}{1-r} \), provided the absolute value of \( r \) is less than 1.

This formula allows the infinite addition of terms to produce a finite number, which might not be intuitive at first, but is a fascinating property of some infinite series.

Convergence of Series

The convergence of a series refers to whether the series approaches a specific value as more terms are added. For geometric series, convergence depends heavily on the common ratio \( r \):

• If \( |r| < 1 \), the series converges, which means there exists a finite limit that the series sums towards.
• Conversely, if \( |r| \geq 1 \), the series diverges, meaning it does not settle towards a specific number.

In the context of the earlier solution, the given geometric series had \( r = \frac{2}{7} \). Since \( \left|\frac{2}{7}\right| < 1 \), the series converges and has a finite sum of \( 22.4 \) after multiplying by the constant factor as derived. The convergence concept plays a critical role in determining the behavior and sum of infinite series.