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Chapter 6: Problem 91

Find the sum of the infinite geometric series. $$\sum_{n=1}^{\infty}(0.1)^{n}$$

Short Answer

Expert verified
The sum of the given infinite geometric series is \(\frac{1}{9}\).

Step by step solution

01

Identify the first term and the common ratio

The given series is \(\sum_{n=1}^{\infty}(0.1)^{n}\). We can observe that the first term \(a\) (when \(n=1\)) of the series is \(0.1\), and the common ratio \(r\) is also \(0.1\).
02

Calculate the sum of the series

The sum \(S\) of an infinite geometric series where \(|r|<1\) can be calculated using the formula \(S = \frac{a}{1 - r}\). Substituting the values of \(a\) and \(r\), we get \(S = \frac{0.1}{1-0.1} = \frac{0.1}{0.9}\).
03

Simplify the result

Upon simplifying the above fraction, we get the result \(S = \frac{1}{9}\).

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Key Concepts

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

First Term in Series
In a geometric series, the first term plays a critical role. It serves as the starting point of the sequence and sets the stage for subsequent terms. In the series \(\sum_{n=1}^{\infty}(0.1)^{n}\), the first term is found by substituting \(n = 1\) into the expression. Therefore, \(a = (0.1)^{1} = 0.1\).
Understanding the first term is essential because it influences the overall sum of the series. When dealing with series problems, always check for the first term first. It forms part of the sum calculation and helps in determining the remaining terms.
  • If the first term is small, each future term will diminish faster.
  • If the first term is large, subsequent values decrease more slowly at first.
Common Ratio in Geometric Series
The common ratio of a geometric series determines how each term relates to the previous one. Simply put, it's the factor by which you multiply each term to get the next. In our series, \(r = 0.1\), meaning each term is 0.1 times the previous term.
To find the common ratio, divide the second term by the first term. For example:
  • The second term: \(0.1 \times 0.1 = 0.01\)
  • The common ratio: \(\frac{0.01}{0.1} = 0.1\)
The value of the common ratio is crucial since it dictates whether the series converges or diverges:
  • If \(|r| < 1)\), the series converges.
  • If \(|r| \geq 1\), there is no finite sum.
Sum of Infinite Series
Calculating the sum of an infinite geometric series can seem daunting, but it's fairly straightforward with the right formula. When the absolute value of the common ratio (\(|r|<1)\) is less than 1, the series has a sum, denoted by \(S\). For our series, the sum is computed as follows:
  • The formula used is \({S = \frac{a}{1 - r}}\).
  • Substitute the first term, \(a = 0.1\), and the common ratio, \(r = 0.1\), to find \(S = \frac{0.1}{1 - 0.1}\).
  • Simplify the expression to compute \(S = \frac{0.1}{0.9} = \frac{1}{9}\).

This result tells us that despite having an infinite number of terms, the entire series converges to the finite value of \(\frac{1}{9}\). Understanding and applying this formula is key because it unveils the total value of an infinite series based on just two parameters鈥� the first term and the common ratio.

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