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Chapter 14: Problem 51

Find the sum of each infinite geometric series. $$\sum_{i=1}^{\infty} 26(-0.3)^{i-1}$$

Short Answer

Expert verified
The sum of the infinite geometric series \(26(-0.3)^{i-1}\) is 20.

Step by step solution

01

Identify the First Term and the Common Ratio

In the series \(26(-0.3)^{i-1}\), the first term \(a\) is the multiplier outside the brackets, which is 26. The common ratio \(r\) is the base of the exponential part which is -0.3.
02

Apply the Infinite Geometric Series Sum Formula

The infinite geometric series sum formula is given by \(S = \frac{a}{1-r}\) where \(S\) is the sum of the series. For this series, the first term \(a\) is 26 and the common ratio \(r\) is -0.3. Therefore, by plugging these values into the formula, we have: \(S = \frac{26}{1-(-0.3)}\).
03

Simplify the Expression

To find the sum \(S\), we can simplify the expression: \(S = \frac{26}{1-(-0.3)}\) becomes \(S = \frac{26}{1.3}\).
04

Calculate the Sum

Now we just need to compute the quotient to find the sum of the series: \(S = \frac{26}{1.3} = 20\)

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Key Concepts

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

Common Ratio
The term "common ratio" is a fundamental concept in understanding geometric series. In a geometric series, each term after the first is found by multiplying the previous term by a fixed number. This fixed number is called the common ratio, denoted by \( r \). For the given series \( 26(-0.3)^{i-1} \), the common ratio \( r \) is \(-0.3\).

Here鈥檚 how it works:
  • Each consecutive term in the series is multiplied by the common ratio \( r \).
  • In our series example, multiplying any term by \(-0.3\) gives the next term.
Common ratios can be positive or negative, and they play a critical role in determining the behavior and convergence of the series. When the absolute value of \( r \) is less than 1, as in this example, the series converges to a specific sum.
First Term
The term "first term" is quite self-explanatory but vital for calculating the sum of a geometric series, especially a geometric series sum. The first term, often denoted by \( a \), is the starting point of the series. In the expression \( 26(-0.3)^{i-1} \), the first term \( a \) is simply the number that is not raised to a power, which is 26.

Knowing the first term is essential because:
  • It sets the initial value from which all other terms in the series are derived by adding products of the common ratio \( r \).
  • It directly influences the overall value of the series' sum.
If you begin your summation calculations without an accurate first term, the resulting sum will be incorrect.
Geometric Series Sum Formula
The geometric series sum formula is paramount for determining the sum of an infinite geometric series, assuming it converges. This formula is given by \( S = \frac{a}{1-r} \), where \( S \) is the sum of the series, \( a \) is the first term, and \( r \) is the common ratio.

For an infinite series to be summed using this formula, two key conditions must be met:
  • The absolute value of the common ratio \( r \) must be less than 1 (i.e., \(|r| < 1\)). This ensures that the series converges.
  • The series must be infinite.
In our exercise, once the values of \( a = 26 \) and \( r = -0.3 \) are known, you can plug these into the formula to find the series sum:
  • Calculate \(1 - (-0.3) = 1.3\).
  • Then, \( S = \frac{26}{1.3} \).
Solving this gives \( S = 20 \), which provides the series' total sum.

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