Problem 35 - 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 \(x .\) $$\sum_{n=1}^{\infty}(-5)^{n} x^{n}$$

Step-by-Step Solution

Answer

The series converges for \(|x| < \frac{1}{5}\) and its sum is \(\frac{-5x}{1+5x}\).

Step 1: Identify the Geometric Series Formula

The series given is \( \sum_{n=1}^{\infty}(-5)^{n} x^{n} \). This is a geometric series of the form \( \sum_{n=1}^{\infty} ar^n \) where \( a = -5x \) and \( r = -5x \).

Step 2: Determine the Convergence Condition

A geometric series \( \sum_{n=1}^{\infty} ar^n \) converges if the absolute value of the common ratio \( r \) is less than 1, i.e., \( |r| < 1 \). For our series, \( r = -5x \). Therefore, the series converges when \(|-5x| < 1\).

Step 3: Solve the Inequality for Convergence

Solve the inequality \( |-5x| < 1 \):1. \( |-5x| < 1 \)2. \( |x| < \frac{1}{5} \)Thus, the series converges when \( |x| < \frac{1}{5} \).

Step 4: Find the Sum of the Convergent Series

When a geometric series converges, it can be summed using the formula \( \frac{a}{1-r} \) where \( a \) is the first term and \( r \) is the common ratio. Here, \( a = -5x \) and \( r = -5x \):\[\text{Sum} = \frac{-5x}{1 - (-5x)} = \frac{-5x}{1 + 5x}\]This is the sum of the series for \( |x| < \frac{1}{5} \).

Key Concepts

Convergence of Series

The concept of series convergence is pivotal in understanding behaviors in infinite series. When we talk about convergence of a series, we mean whether the sum of its infinitely many terms approaches a specific, finite value. If a series converges, it means as you keep adding more terms, the total gets closer to some number rather than growing endlessly or oscillating wildly. For geometric series specifically, their convergence depends largely on the value of the common ratio.

For a geometric series of the form \( \sum_{n=1}^{\infty} ar^n \), this infinite series converges when the absolute value of the common ratio \( r \) is less than 1 or \( |r| < 1 \). In our example with the series \( \sum_{n=1}^{\infty}(-5)^n x^n \), the expression \( -5x \) acts as the common ratio. Therefore:

• Calculate \( |-5x| < 1 \).
• Solve the inequality to find \(|x| < \frac{1}{5}\).

This means that the series converges only when \( x \) values fall between \( -\frac{1}{5} \) and \( \frac{1}{5} \).
Understanding this concept helps in predicting when a series will add up to a sensible number.

Sum of Geometric Series

Once we know a geometric series converges, the next natural question is: "What is the sum of this series?" This is particularly relevant when the goal is not just to determine whether the series converges, but also to find a tangible value that represents its sum.

For any converging geometric series \( \sum_{n=1}^{\infty} ar^n \), the sum can be calculated using the formula:\[\text{Sum} = \frac{a}{1-r}\]where \(a\) is the first term and \( r \) is the common ratio. For the exercise given, \( a = -5x \) and \( r = -5x \). Plug these into the formula:

• First term \(a = -5x\), common ratio \(r = -5x\).
• Use the sum formula \(\frac{-5x}{1+5x}\).

The sum \( \frac{-5x}{1+5x} \) tells us the aggregate value of the series for each viable \( x \) within the convergence range.
This calculation assumes \(|x| < \frac{1}{5}\). It’s important to get this right, to ensure we're only summing when it's valid.

Geometric Series Formula

A geometric series is a series of numbers with each term found by multiplying the previous one by a fixed, non-zero number called the "common ratio". This structured pattern allows us to derive a formula for the sum of convergent series. The general formula for the sum of an infinite geometric series \( \sum_{n=1}^{\infty} ar^n \) is given by\[\text{Sum} = \frac{a}{1 - r}\]where:

• \(a\) is the first term.
• \(r\) is the common ratio, and its absolute value must be less than 1 for convergence.

This formula is incredibly useful because it provides a single expression to capture the sum rather than manually adding each term. In the example with series \( \sum_{n=1}^{\infty} (-5)^n x^n \), our formula becomes extremely practical. With \( a = -5x \) and \( r = -5x \), the sum is straightforwardly \( \frac{-5x}{1 + 5x} \).
Knowing this formula empowers students to handle similar problems, simplifying what might seem like an overwhelming calculation into a more manageable approach.