Problem 113 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the indicated partial sum of the series. \(\sum_{i=1}^{\infty} \frac{7}{5^{i}}\) Fourth partial sum

Step-by-Step Solution

Answer

The fourth partial sum of the series is \(\frac{343}{125}\)

Step 1: Identify a, r and n

First we need to identify the first term (a), the common ratio (r), and the number of terms (n). Here, the first term a is \(\frac{7}{5}\), the common ratio r is \(\frac{1}{5}\) and the number of terms n is 4 (since it's the fourth partial sum).

Step 2: Calculating the Partial Sum

After identifying a, r, and n, we plug these values in the formula for the partial sum. This gives us \[S = \frac{7/5*(1-(1/5)^4)}{1-1/5}\]

Step 3: Simplify the Expression

We simplify the expression in the numerator and denominator separately: \[S = \frac{7/5*(1-1/625)}{4/5}\]. Further simplification and arithmetic calculation lead to: \[S = \frac{343}{125}\].

Key Concepts

Partial Sums

When dealing with a geometric series, the concept of partial sums is essential to understand. Partial sums represent the sum of the first few terms in the series. For instance, in the given problem, we are tasked with finding the fourth partial sum. This means we need to add up the first four terms of the series.

The formula to calculate a partial sum of a geometric series is:

• \( S_n = a \frac{1-r^n}{1-r} \), where:
• \( S_n \) is the partial sum of the first \( n \) terms,
• \( a \) is the first term,
• \( r \) is the common ratio,
• and \( n \) is the number of terms to sum.

Understanding partial sums helps you handle series without calculating each term individually. Instead, you can use this formula to simplify your work and find the sum quickly.

Common Ratio

The common ratio is a key component in understanding a geometric series. It tells you how each term in the series changes from the previous term. In our example, the common ratio (\( r \)) is \( \frac{1}{5} \). This means each term is found by multiplying the preceding term by \( \frac{1}{5} \).

To identify the common ratio, take any term in the series and divide it by the term before it. This should be constant across the series. Recognizing this pattern helps in calculating sums and assessing the behavior of the series over a long range.

• A common ratio greater than 1 indicates the series will grow.
• If it is between 0 and 1, the series will shrink.
• A negative ratio may cause the series to alternate in sign.

Arithmetic Calculations

Arithmetic calculations are necessary when solving problems related to geometric series. After identifying the needed components like the first term, common ratio, and number of terms, it is time to perform calculations.

For our specific series, we computed the fourth partial sum using the formula mentioned earlier. We substituted the first term, common ratio, and number of terms into the equation: \( S = \frac{\frac{7}{5} (1-(\frac{1}{5})^4)}{1-\frac{1}{5}} \).

Next, each part of the equation was simplified:

• The power \( (\frac{1}{5})^4 \) was calculated,
• then subtracted from 1,
• and multiplied by the first term,
• finally, the entire expression was divided by \( (1-\frac{1}{5}) \).

This series of arithmetic calculations ultimately results in the partial sum \( \frac{343}{125} \). Correct arithmetic is crucial to obtaining the right result.

Infinite Series

In an infinite series, terms extend endlessly. However, it is crucial to realize that while the number of terms is infinite, sometimes the series converges to a specific number. This depends significantly on the common ratio.

In the series given, while we are only examining a few terms to find the partial sum, the implications of extending the series infinitely can be quite different. If the common ratio \( |r| < 1 \), the series can converge to a finite sum.
To determine this, we use the formula for a **geometric series** sum:

• \( S = \frac{a}{1-r} \), which is employed when dealing with an infinite series.

This concept helps in understanding how forces like interest or decay work over time, mathematically modeling them through an infinite series.