/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 Evaluate \(\sum_{m=2}^{\infty} \... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 17

Evaluate \(\sum_{m=2}^{\infty} \frac{5}{6^{m}}\)

Short Answer

Expert verified
The given infinite geometric series is \(\sum_{m=2}^{\infty} \frac{5}{6^{m}}\), with the first term \(a = \frac{5}{6^2}\) and common ratio \(r = \frac{1}{6}\). Using the formula for the sum of an infinite geometric series, \(S = \frac{a}{1 - r}\), we find that the sum of the series is 1.

Step by step solution

01

Identify the first term and common ratio

The given series is \(\sum_{m=2}^{\infty} \frac{5}{6^{m}}\). Let's identify the first term 'a' and the common ratio 'r'. The first term 'a' can be found by plugging m = 2 into the given series: \(a = \frac{5}{6^2}\). The common ratio 'r' is the factor by which consecutive terms change, which is \(\frac{1}{6}\) in this case.
02

Apply the formula for the sum of an infinite geometric series

Now that we have identified the first term and common ratio, we can apply the formula for the sum of an infinite geometric series: \(S = \frac{a}{1 - r}\). We can plug in the values of 'a' and 'r' we found earlier: \(S = \frac{\frac{5}{6^2}}{1 - \frac{1}{6}}\).
03

Simplify the expression

Let's simplify the expression by first simplifying the denominator: \(1 - \frac{1}{6} = \frac{5}{6}\). Now we can substitute this back into the expression for S: \(S = \frac{\frac{5}{6^2}}{\frac{5}{6}}\). To further simplify this, we will multiply both numerator and denominator by \(6^2\): \(S = \frac{5}{5} = 1\). So, the sum of the given infinite geometric series is 1.

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Key Concepts

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

Infinite Series
An infinite series is a sum of an infinite number of terms. Imagine lining up an endless row of numbers and adding them up鈥攖his is the heart of an infinite series. In mathematics, it's important to understand not all infinite series have a definite sum; some just keep growing as you add more terms. However, under certain conditions, such as when a series is 'convergent', we can talk about its sum even though there are infinitely many terms. The sum of certain infinite series can be finite and very much meaningful, providing profound implications in fields like physics, engineering, and finance.
Geometric Series
A geometric series is a type of series where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the 'common ratio'. This pattern can be super helpful for calculating quantities that grow or shrink exponentially, such as interest in a bank or the decay of a radioactive substance. The beauty of geometric series lies in their predictability鈥攐nce you know the first term and the common ratio, you can determine any term in the series or even the sum of the series if it鈥檚 infinite and converges.
Convergence of Series
The convergence of a series addresses whether the sum of its terms approaches a specific value or not. For a series to converge, as we add more and more terms, they should get smaller and closer to zero quickly enough that the total doesn't just shoot off to infinity. A classic example of convergence is seen in the Zeno's paradox, where each term is half the previous one鈥攂y adding up the infinite terms, we get a finite distance. Convergence is a cornerstone in understanding infinite series since it tells us if it makes sense to talk about their sum.
Sum of Series
Determining the sum of a series involves adding up all its terms. While this sounds straightforward for a finite number of terms, it gets more nuanced with an infinite series. The sum of an infinite series, when it exists, is a limit that the series approaches as the number of terms grows without bound. This concept comes in handy when dealing with repeating patterns or processes without a distinct end. Knowing how to find the sum of infinite series is essential for solving real-world problems where processes continue indefinitely.
Sequences and Series
Sequences and series are fundamental concepts in mathematics that describe ordered lists of numbers. A sequence is simply the list of numbers in a specific order, and a series is the sum of the terms of a sequence. While the terms sequence and series are sometimes used interchangeably, it's essential to distinguish between them: a sequence refers to the ordered list, and a series refers to the summing-up of the elements of the sequence. Both play starring roles in various scenarios, from calculating loan payments to predicting population growth.

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Most popular questions from this chapter

In the decimal expansion of \(0.9^{9999}\), how many zeros follow the decimal point before the first nonzero digit?

Consider an arithmetic sequence with first term b and difference \(d\) between consecutive terms. (a) Write the sequence using the three-dot notation, giving the first four terms of the sequence. (b) Give the \(100^{\text {th }}\) term of the sequence. \(b=8, d=-5\)

Explain why 0.2 and the repeating decimal \(0.199999 \ldots\) both represent the real number \(\frac{1}{5}\).

Show that the sum of a finite arithmetic se- \(-\) quence is 0 if and only if the last term equals the negative of the first term.

In Exercises \(31-34,\) write the series using summation notation (starting with \(m=1\) ). Each series in Exercises \(31-34\) is either an arithmetic series or \(a\) geometric series. \(\frac{5}{9}+\frac{5}{27}+\frac{5}{81}+\cdots+\frac{5}{3^{40}}\)
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