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Chapter 9: Problem 76

Find the sum of the infinite geometric series. $$\sum_{n=0}^{\infty} 4(0.2)^{n}$$

Short Answer

Expert verified
The sum of the given infinite geometric series is 5.

Step by step solution

01

Identify the first term (\a\) and the common ratio (\r\)

In this series, the first term is \( a = 4(0.2)^0 = 4 \) and the common ratio is \( r = 0.2 \) as it's the base of the exponent in the formula.
02

Check |r| < 1

In order for the geometric series to converge and its sum to be finite, the absolute value of the common ratio must be less than 1, i.e., \( |r| < 1 \). In this case, \( |0.2| < 1 \), so the series is convergent.
03

Apply the formula to find the sum

The sum \( S \) of an infinite geometric series can be found by applying the formula \( S = \frac{a}{1 - r} \). Substituting the values we have: \( S = \frac{4}{1 - 0.2} = 5 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

geometric series sum formula
A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. To find the sum of an infinite geometric series, we utilize the geometric series sum formula.The formula is given by: \[ S = \frac{a}{1 - r} \]This formula is applicable under the condition that the series converges, which occurs when the absolute value of the common ratio \( r \) is less than 1 (\( |r| < 1 \)). Here's a simple breakdown:
  • \( S \) represents the sum of the infinite series.
  • \( a \) is the first term of the series.
  • \( r \) is the common ratio.
Whenever these values are identified, you can substitute them into the formula to find the sum.
convergence of series
For an infinite geometric series to have a sum, it must converge. This means that as you add more terms, the series approaches a finite number. The crucial factor determining convergence is the common ratio.A series converges if the absolute value of the common ratio is less than 1 (\( |r| < 1 \)). This essentially means that the terms are getting smaller and smaller, leading the partial sums to get closer and closer to a certain value. If \(|r|\) is greater than or equal to 1, the series will diverge, meaning the sum doesn't converge to a particular value.In the given exercise, where \( r = 0.2 \), the absolute value is \( |0.2| < 1 \), so the series converges. This condition allows us to confidently apply the formula for the sum of the series.
first term and common ratio
In a geometric series, the first term (denoted by \( a \)) and the common ratio (denoted by \( r \)) are foundational elements.- **First Term \( a \):** This is simply the starting point of the series. It's the initial term from which all subsequent terms are derived. - In our example, the first term \( a = 4 \), because \( 4(0.2)^0 = 4 \).- **Common Ratio \( r \):** This is the factor by which each term is multiplied to get the next term in the series. - In the exercise, \( r = 0.2 \). Each term is multiplied by this ratio to produce the next term.Knowing these two components is essential for applying the geometric series sum formula and checking for convergence. They are the keys to figuring out the series sum as long as the series converges.

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Most popular questions from this chapter

Use the Binomial Theorem to expand the complex number. Simplify your result. $$\left(-\frac{1}{2}+\frac{\sqrt{3}}{2} i\right)^{3}$$

Use the Binomial Theorem to expand the complex number. Simplify your result. $$(2-i)^{5}$$

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. What is the relationship between the two graphs? Use the Binomial Theorem to write the polynomial function \(g\) in standard form.$$f(x)=-x^{4}+4 x^{2}-1, \quad g(x)=f(x-3)$$.

Complete each expression for the apparent \(n\) th term \(a_{n}\) of the sequence. Which expressions are appropriate to represent the cost \(a_{n}\) to buy \(n\) MP3 songs at a cost of \(\$ 1\) per song? Explain. $$\text { (a) } a_{n}=1 \square$$ $$\text { (b) } a_{n}=\frac{ \square 1}{(n-1) !}$$ $$\text { (c) } a_{n}=\sum_{k=1}^{n}$

Solve for \(n\) $$4 \cdot_{n+1} P_{2}=_{n+2} P_{3}$$
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