Problem 72 - A Graphical Approach to Precalculus with Limits

Question

Determine the largest value of \(n\) that satisfies the inequality. $$\sum_{k=1}^{n} \frac{2}{5^{k}}<\frac{1}{2}$$

Step-by-Step Solution

Answer

The largest value of \(n\) is 2.

Step 1: Understand the Problem

The problem requires finding the largest integer \( n \) such that the sum of a geometric series is less than \( \frac{1}{2} \). The series is \( \sum_{k=1}^{n} \frac{2}{5^k} \).

Step 2: Identify the Type of Series

We recognize this as a geometric series with first term \( a = \frac{2}{5} \) and common ratio \( r = \frac{1}{5} \).

Step 3: Use the Sum Formula for a Geometric Series

The sum of the first \( n \) terms of a geometric series is given by the formula \( S_n = a \frac{1 - r^n}{1 - r} \). Substitute \( a = \frac{2}{5} \) and \( r = \frac{1}{5} \) into the formula, so \[ S_n = \frac{2}{5} \cdot \frac{1 - (\frac{1}{5})^n}{1 - \frac{1}{5}} \].

Step 4: Simplify the Sum Formula

Simplify \( S_n \) by computing \( 1 - \frac{1}{5} = \frac{4}{5} \), leading to \[ S_n = \frac{2}{5} \cdot \frac{1 - (\frac{1}{5})^n}{\frac{4}{5}} = \frac{1}{2} (1 - (\frac{1}{5})^n) \].

Step 5: Set the Inequality

We set our inequality based on the requirement of the problem: \[ \frac{1}{2}(1 - (\frac{1}{5})^n) < \frac{1}{2} \].

Step 6: Solve the Inequality

Cancel the \( \frac{1}{2} \) from both sides to get \[ 1 - \left(\frac{1}{5}\right)^n < 1 \]. This simplifies to \[ \left(\frac{1}{5}\right)^n > 0 \] which holds for all \( n \). Continue by solving \[ \left(\frac{1}{5}\right)^n < \frac{1}{2} \].

Step 7: Evaluate for Largest \(n\)

Convert to logarithmic form to find \( n \): \[ n \log (\frac{1}{5}) < \log(\frac{1}{2}) \]. Since \( \log(\frac{1}{5}) < 0 \), multiply through by \( -1 \) and reverse the inequality:\[ n > \frac{\log(\frac{1}{2})}{\log(\frac{1}{5})} \approx 2.32 \].

Step 8: Find Integer \(n\)

The largest integer \( n \) less than or equal to 2.32 is \( n = 2 \). Check for \( n = 2 \) with \( S_2 = \frac{1}{2}(1 - \frac{1}{25}) = \frac{1}{2}(\frac{24}{25}) = \frac{12}{25} = 0.48 < 0.5 \), and check for \( n = 3 \) with \( S_3 = \frac{1}{2}(\frac{124}{125}) = 0.496 > 0.5 \). So, \( n = 2 \) is verified.

Key Concepts

Series Convergence

In mathematics, a geometric series is a series with a constant ratio between successive terms. A key concept related to this is series convergence, which determines whether the series approaches a specific value as the number of terms tends towards infinity.

The geometric series in our problem is given by \( \sum_{k=1}^{n} \frac{2}{5^k} \). This series will converge if the absolute value of the common ratio \( r \) is less than 1. In this case, our common ratio \( r \) is \( \frac{1}{5} \), which satisfies the convergence condition because it is indeed less than 1.

Convergence signifies that as \( n \) increases, the sum approaches a particular limit. For our geometric series, the formula for the sum of the first \( n \) terms is \( S_n = a \frac{1 - r^n}{1 - r} \) where \( a \) is the first term and \( r \) is the common ratio. Understanding this convergence helps in calculating how large \( n \) can be while satisfying the inequality condition.

Inequality Solving

Solving inequalities is an important part of mathematical problem-solving, especially when determining specific constraints, like finding the maximum value of \( n \) in our series problem.

To solve the inequality \( S_n < \frac{1}{2} \), where \( S_n = \frac{1}{2}(1 - (\frac{1}{5})^n) \), we first eliminate constants or simplifying factors that make it easier to manage. By removing the constant \( \frac{1}{2} \) from both sides of our problem, the inequality simplifies to \( (1 - (\frac{1}{5})^n) < 1 \). This simplifies further to \( (\frac{1}{5})^n > 0 \).

This simplification is crucial as it establishes the problem's core constraint: ensuring the series sum using a finite \( n \) remains less than \( \frac{1}{2} \). At this stage, translating these inequalities into a logarithmic format allows us to solve for \( n \).

Solving inequalities involves manipulating the expression until \( n \) values that meet the condition are clear, like realizing that \( n = 2 \) satisfies \( (\frac{1}{5})^n < \frac{1}{2} \) but \( n = 3 \) does not.

Logarithmic Inequality

Logarithmic inequalities are a powerful tool for solving exponential inequalities, such as those that appear in geometric series problems.

In this problem, the inequality \( (\frac{1}{5})^n < \frac{1}{2} \) was tackled using logarithms. By converting exponential inequalities into a logarithmic form, they become linear, making them simpler to solve.

Taking the logarithm of both sides gives \( n \log(\frac{1}{5}) < \log (\frac{1}{2}) \). Here, understanding that logarithms of numbers less than 1 are negative is key. Therefore, multiplying by \(-1\) reverses the inequality direction, resulting in \( n > \frac{\log(\frac{1}{2})}{\log(\frac{1}{5})} \).

This transformation was critical to finding that \( n \approx 2.32 \), which when rounded down due to the series condition, determines that \( n = 2 \) is the largest integer satisfying our initial inequality. This demonstrates how logarithmic manipulations can solve complex inequality problems effectively.