Problem 76 - Thomas Calculus

Question

In Exercises \(73-78,\) find the values of \(x\) for which the given geometric series converges. Also, find the sum of the series (as a function of \(x )\) for those values of \(x .\) $$ \sum_{n=0}^{\infty}\left(-\frac{1}{2}\right)^{n}(x-3)^{n} $$

Step-by-Step Solution

Answer

The series converges for \(1 < x < 5\), with the sum as \( S(x) = \frac{2}{5-x} \).

Step 1: Identify the Common Ratio

For the series \( \sum_{n=0}^{\infty} (-\frac{1}{2})^n (x-3)^n \), the common ratio is \( r = -\frac{1}{2}(x-3) \).

Step 2: Determine Convergence Condition

For an infinite geometric series to converge, the absolute value of the common ratio must be less than 1. Therefore, we have \(| -\frac{1}{2}(x-3) | < 1\).

Step 3: Solve the Inequality for Convergence

Simplifying \(| -\frac{1}{2}(x-3) | < 1\), we get two inequalities: \[ -1 < -\frac{1}{2}(x-3) < 1 \]. Solving these inequalities, For \(-\frac{1}{2}(x-3) > -1\):\[ x - 3 < 2 \]\[ x < 5 \].For \(-\frac{1}{2}(x-3) < 1\):\[ x - 3 > -2 \]\[ x > 1 \].

Step 4: Combine the Inequalities

The inequalities \( x > 1 \) and \( x < 5 \) combine to give the interval \( 1 < x < 5 \). This is the range of \( x \) for which the series converges.

Step 5: Find the Sum of the Convergent Series

For a convergent geometric series, the sum \( S \) is given by the formula:\[ S = \frac{a}{1 - r} \], where \( a \) is the first term of the series. Here, \( a = 1 \) and \( r = -\frac{1}{2}(x-3) \). Thus, the sum is:\[ S(x) = \frac{1}{1 + \frac{1}{2}(x-3)} \] Simplifying gives:\[ S(x) = \frac{2}{5-x} \]for \( 1 < x < 5 \).

Key Concepts

Common Ratio

In a geometric series, the **common ratio** is a crucial component that links consecutive terms. It tells us the factor by which each term in the series is multiplied to get the next term. Understanding the common ratio helps determine if a series converges.

For the given series, the common ratio is identified from the series \( \sum_{n=0}^{\infty} \left(-\frac{1}{2}\right)^{n}(x-3)^{n} \). Here, the common ratio \( r \) is \(-\frac{1}{2}(x-3)\). It is formed by multiplying each term by this ratio to produce the next term.

The value of this ratio can change based on what's inside the parentheses, thus allowing flexibility in understanding different values of \( x \). This variability is key in solving for convergence.

Series Convergence Criteria

The **convergence criteria** for geometric series is what allows us to determine if an infinite series will sum to a finite value. This is crucial because infinite series can otherwise go on forever and not be calculable.

The primary criteria for a geometric series to converge is that the absolute value of the common ratio \( |r| \) must be less than 1. Mathematically speaking, this is written as \( | -\frac{1}{2}(x-3) | < 1 \).

When dealing with series convergence, think of this condition as a balancing act. If \( r \) is too large or doesn't satisfy this inequality, the series is not balanced and won’t converge.

Inequality Solving

**Inequality solving** is an essential skill to figure out for what values of \( x \) the series will converge. It requires a series of steps where we break down the inequality into simpler parts.

Given the inequality \(-1 < -\frac{1}{2}(x-3) < 1\), we need to solve it by considering both sides separately:

• First, solving \(-\frac{1}{2}(x-3) > -1\) gives \(x-3 < 2\), hence \(x < 5\).
• Secondly, solving \(-\frac{1}{2}(x-3) < 1\) gives \(x-3 > -2\), thus \(x > 1\).

Putting these results together, we find that the values that satisfy both conditions are within the interval \(1 < x < 5\). This tells us exactly which values make the series converge.

Sum of Series Formula

Once the series convergence is confirmed, the next crucial step is finding the **sum of the series**.

For geometric series, the sum formula is \( S = \frac{a}{1 - r} \)where \( a \) is the first term of the series and \( r \) is the common ratio.

In this instance, with \( a = 1 \) and \( r = -\frac{1}{2}(x-3) \), the sum becomes \( S(x) = \frac{1}{1 + \frac{1}{2}(x-3)} \).

By simplifying, the sum for this convergent series is \( S(x) = \frac{2}{5-x} \)for the interval \( 1 < x < 5 \). This provides a clear function depending on \( x \) that gives the sum of the convergent series, tying together our understanding of the series behavior.