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Chapter 18: Problem 7

Write the sum using summation notation. (a) \(\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\cdots+\frac{2}{3^{n}}+\cdots\) (b) \(\frac{2}{3}+2+6+\cdots+2(3)^{n}+\cdots\)

Short Answer

Expert verified
(a) The summation notation for the series is \(\sum_{n=1}^{\infty} \frac{2}{3^{n}}\). (b) The summation notation for the series is \(\sum_{n=-1}^{\infty} 2(3^{n})\)

Step by step solution

01

Observe the Pattern in Series (a)

In the series \(\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\cdots+\frac{2}{3^{n}}+\cdots\), it can be observed that each term is a fraction where the numerator is constant, 2 and the denominator is a power of 3.
02

Formulate the General Term (Series a)

Based on the pattern observed in the series, the general term for (a) can be expressed as \(\frac{2}{3^{n}}\), where \(n\) starts at 1 and increases by 1.
03

Write the Summation Notation (Series a)

Using the sigma notation for series, the series (a) can be represented as \(\sum_{n=1}^{\infty} \frac{2}{3^{n}}\)
04

Observe the Pattern in Series (b)

In the series \(\frac{2}{3}+2+6+\cdots+2(3)^{n}+\cdots\), it can be observed that each term is increasing power of 3 starting at negative 1 and the entire term is multiplied by 2.
05

Formulate the General Term (Series b)

Based on the pattern observed in the series, the general term for (b) can be expressed as \(2(3^{n})\), where \(n\) starts at -1 and increases by 1.
06

Write the Summation Notation (Series b)

Using the sigma notation for series, the series (b) can be represented as \(\sum_{n=-1}^{\infty} 2(3^{n})\)

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Key Concepts

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

Geometric Series
A geometric series is a sum of terms in a sequence where each term is found by multiplying the previous term by a constant factor. This constant is known as the common ratio. Understanding geometric series is crucial when evaluating many types of sequences.
  • For example, in series (a) from the exercise, we have \((\frac{2}{3}) + (\frac{2}{9}) + (\frac{2}{27}) + \cdots\),\ where each term is formed by multiplying the preceding term by \(\frac{1}{3}\).
  • In series (b), the sequence \((\frac{2}{3}) + 2 + 6 + \cdots\) contains terms that are each a multiplied result by a power of 3, denoted as \(2(3^{n})\).
  • The importance of the geometric series is its ability to model exponential growth or decay in various scenarios.
A key feature of geometric series is the fact they can sum up to a finite value if the common ratio is between -1 and 1, even if they have infinitely many terms. This is the basis for the concept called a converging series.
Infinite Series
An infinite series is a series that continues indefinitely, without stopping. This is represented mathematically by extending the upper limit of the sum to infinity. In an infinite series, one challenges understanding because the addition of infinitely many terms might not converge to a finite sum.
  • In exercise (a), the notation \(\sum_{n=1}^{\infty}\frac{2}{3^n}\) signifies an infinite series due to the upper limit being infinity.
  • Similarly, series (b) \(\sum_{n=-1}^{\infty} 2(3^n)\) also represents an infinite series.
  • The concept of convergence is significant; it refers to the series approaching a fixed value as more terms are added.
To determine whether a series like these converges or not, it may require specific tests or analyses, particularly useful in the study of calculus and mathematical sequences.
Sigma Notation
Sigma notation is a compact and formal way to express the sum of terms. The symbol used is \(\Sigma\), which is the Greek letter sigma, standing for "sum."
  • This notation highlights the starting and ending points of summation, as well as the general term to sum.
  • For instance, in series (a), \(\sum_{n=1}^{\infty}\frac{2}{3^n}\), indicates adding terms starting from \(n=1\) through to infinity.
  • In series (b), \(\sum_{n=-1}^{\infty} 2(3^n)\), reflects a summation starting with \(n=-1\).
Sigma notation serves as a powerful tool because it provides a clear, concise way to deal with complex sequences and sums, often encountered in mathematics and its applications.

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Most popular questions from this chapter

People who have slow metabolism due to a malfunctioning thyroid can take thyroid medication to alleviate their condition. For example, the boxer Muhammad Ali took Thyrolar 3, which is 3 grains of thyroid medication, every day. The amount of the drug in the bloodstream decays exponentially with time. The half-life of Thyrolar is 1 week. (a) Suppose one 3 -grain pill of Thyrolar is taken. Write an equation for the amount of the drug in the bloodstream \(t\) days after it has been taken. (Hint: In part (a) you are dealing with one 3 -grain pill of Thyrolar. Knowing the half-life of Thyrolar, you are asked to come up with a decay equation. This part of the problem has nothing to do with geometric sums.) (b) Suppose that Ali starts with none of the drug in his bloodstream. If he takes 3 grains of Thyrolar every day for ve days, how much Thyrolar is in his bloodstream immediately after having taken the fth pill? (c) Suppose Ali takes 3 grains of Thyrolar each day for one month. How much thyroid medication will be in his bloodstream right before he takes his 31 st pill? Right after? (d) After taking this medicine for many years, what was the amount of the drug in his body immediately after taking a pill? Historical note: Before one of his last ghts Muhammad Ali decided to up his dosage to 6 grains. In doing so he mimicked the symptoms of an overactive thyroid. The result in terms of the ght was dismal.

You have found a calling! You have some burning questions about elephants and want desperately to go to Kenya for a year. In addition to the plane fare you ll need some equipment, a guide, a jeep \(\ldots\) You ll need some money. You gure that you 11 need \(\$ 7000 .\) Each month beginning today you plan to put a xed amount of money into an account paying \(6 \%\) interest compounded monthly. How much must you deposit into the account each month if you plan to begin your eld work in four years?

For Problems 1 through 10, write the sum using summation notation. $$ 2^{3}+3^{3}+4^{3}+\cdots+19^{3} $$

For Problems , determine whether the series converges or diverges. Explain your reasoning. $$ \frac{1}{1000}+\frac{2}{1000}+\frac{3}{1000}+\frac{4}{1000}+\cdots $$

Determine whether the series converges or diverges. If it converges, \(n d\) its sum. $$ (2 e)^{-2}+(2 e)^{-3}+(2 e)^{-4}+\cdots+(2 e)^{-n}+\cdots $$
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