Problem 36 - Essential Calculus Early Transcendentals

Question

Find the values of \(x\) for which the series converges. Find the sum of the series for those values of \(x .\) $$\sum_{n=0}^{\infty}(-4)^{n}(x-5)^{n}$$

Step-by-Step Solution

Answer

Converges when \( 4.75 < x < 5.25 \); sum is \( \frac{1}{1 + 4(x-5)} \).

Step 1: Identify the Series

The given series is \( \sum_{n=0}^{\infty}a_n \) where \( a_n = (-4)^n (x-5)^n \). This resembles a geometric series with the general term \( a_n = ar^n \) where \( a = 1 \) and \( r = (-4)(x-5) \).

Step 2: Find Convergence Condition

For an infinite geometric series \( \sum_{n=0}^{\infty} ar^n \) to converge, the common ratio \( r \) must satisfy \( |r| < 1 \). Here, \( r = (-4)(x-5) \). Therefore, we need to solve \( |-4(x-5)| < 1 \).

Step 3: Solve the Inequality

The inequality \( |-4(x-5)| < 1 \) can be simplified and solved as:\[ |-4(x-5)| = |(-4)(x-5)| = 4|x-5| < 1 \]\[ |x-5| < \frac{1}{4} \]Thus, the values of \( x \) must satisfy \( 5 - \frac{1}{4} < x < 5 + \frac{1}{4} \).This simplifies to the interval \( 4.75 < x < 5.25 \).

Step 4: Find the Sum of the Series

If the geometric series \( \sum_{n=0}^{\infty} ar^n \) converges, its sum is given by \( \frac{a}{1-r} \). Since \( a = 1 \) and \( r = -4(x-5) \), the sum of the series is:\[ S = \frac{1}{1 - (-4)(x-5)} = \frac{1}{1 + 4(x-5)} \].

Step 5: Substitute Valid \(x\) Values Into the Sum

The sum formula \( S = \frac{1}{1 + 4(x-5)} \) is valid for \( 4.75 < x < 5.25 \). For any \( x \) in this interval, substitute \( x \) into the sum expression to get the sum of the series.

Key Concepts

Series Convergence

Convergence of a series is crucial as it tells us whether the series sums up to a finite value. In the context of a geometric series, convergence happens under specific conditions. For a series like \[ \sum_{n=0}^{\infty} ar^n \] the essential condition for convergence is that the absolute value of the common ratio \(r\) should be less than 1: \[ |r| < 1 \]This means the terms must shrink towards zero as the series extends to infinity. For the series in the original exercise, \[ r = -4(x-5) \] The task was to determine the values of \(x\) for which the series converges. Solving \[ |-4(x-5)| < 1 \] revealed that the series converges when \(x\) falls between 4.75 and 5.25.

Sum of Series

Once it's determined that a series converges, we can calculate its sum if a formula exists. For convergent geometric series, the sum is given by:\[ S = \frac{a}{1-r} \]where \(a\) is the first term and \(r\) is the common ratio. In the problem, \(a = 1\) and \(r = -4(x-5)\), so the sum becomes:\[ S = \frac{1}{1 + 4(x-5)} \]This formula helps calculate the series' sum for any \(x\) in the convergence range \(4.75 < x < 5.25\). Simply plug in the value of \(x\) from the interval into the expression to find the sum.

Inequalities

Inequalities are mathematical expressions showing the relationship between two values when they are not equal. Solving an inequality involves finding the set of values which make the inequality true. The inequality \[ |-4(x-5)| < 1 \] was crucial in the series convergence exercise. Its solution meant breaking it down to \[ 4|x-5| < 1 \]and then to \[ |x-5| < \frac{1}{4} \]This explains that \(x\) needed to lie within a specific range about the number 5. In this case, the solution revealed:

• \(x > 4.75\)
• \(x < 5.25\)

This gave us the values of \(x\) for which the series would converge.

Infinite Series

Infinite series, as the name suggests, have an unlimited number of terms. They play a pivotal role in mathematics, especially in calculus and analysis, often representing phenomena that extend indefinitely. Unlike finite sums, infinite series can still result in finite values if they converge. A noteworthy aspect of such series is their potential to sum to a finite number even with infinite terms, provided they decrease adequately fast.
Geometric series are a key type of infinite series. When each term is a constant multiple of the previous term, convergence and sum formulas can potentially apply. For the series in the given example, determining convergence and then using the sum formula is only possible with infinite terms aligning to these constraints, showing the elegant blend of infinity with finite outcomes in mathematics.