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Chapter 8: Problem 43

Find the sum of each infinite geometric series. $$\sum_{i=1}^{\infty} 8(-0.3)^{i-1}$$

Short Answer

Expert verified
The sum of the infinite geometric series \(\sum_{i=1}^{\infty} 8(-0.3)^{i-1}\) is approximately 6.15.

Step by step solution

01

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

The first term (a) is the coefficient of the series which is 8. The common ratio (r) is the base of the exponent, in this case -0.3. So, we have a=8 and r=-0.3
02

Check if series converges

The series will converge if -1< r< 1. In this case the common ratio r=-0.3 lies between -1 and 1, so the series converges and we can find the sum.
03

Apply the sum formula

Applying the formula \(S = \frac{a}{1-r}\), with a=8 and r=-0.3, we get \(S = \frac{8}{1-(-0.3)}\)
04

Simplify the equation

We have \(S = \frac{8}{1.3}\) after performing the subtraction in the denominator.
05

Calculate the sum

Upon performing the division, we get \(S \approx 6.15\)

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Most popular questions from this chapter

Explain how to use the Binomial Theorem to expand a binomial. Provide an example with your explanation.

How do you determine how many terms there are in a binomial expansion?

Evaluate the given binomial coefficient. $$ \left(\begin{array}{c}100 \\\98\end{array}\right) $$

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ (x+2)^{8} $$

Which one of the following is true? a. The sequence \(2,6,24,120, \ldots\) is an example of a geometric sequence. b. The sum of the geometric series \(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots+\frac{1}{512}\) can only be estimated without knowing precisely which terms occur between \(\frac{1}{8}\) and \(\frac{1}{512}\). c. \(10-5+\frac{5}{2}-\frac{5}{4}+\cdots=\frac{10}{1-\frac{1}{2}}\) d. If the \(n\) th term of a geometric sequence is \(a_{n}=3(0.5)^{n-1},\) the common ratio is \(\frac{1}{2}\).
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