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Chapter 10: Problem 37

Determine the convergence or divergence of the series. Use a symbolic algebra utility to verify your result. $$ \sum_{n=0}^{\infty}(1.075)^{n} $$

Short Answer

Expert verified
The series \( \sum_{n=0}^{\infty}(1.075)^{n} \) diverges.

Step by step solution

01

Identify the ratio

The ratio of a geometric series is the constant factor that each term is multiplied by to get the next term. Looking at the series \( \sum_{n=0}^{\infty}(1.075)^{n} \), it's clear that the ratio \(r\) is \(1.075.\)
02

Determine if the series converges

For a geometric series to converge, the absolute value of the ratio must be less than 1. In our case, \(|1.075|\) is not less than 1. Thus, we can conclude that the series does not converge.
03

Verify with a symbolic algebra utility

Using a symbolic algebra utility like Wolfram Alpha, inputting the series \( \sum_{n=0}^{\infty}(1.075)^{n} \) should confirm that the series indeed diverges.

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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 sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number. This fixed number is referred to as the "common ratio." It's one of the simplest types of series in mathematics and is broadly used in various fields.

### Characteristics of a Geometric Series
  • It has a first term, often called "a," and a common ratio, "r."
  • The nth term can be written as: \(a \cdot r^{n-1}\).
The infinite series is expressed as: \[\sum_{n=0}^{\infty} a \cdot r^{n}\] When analyzing a geometric series, one key aspect is whether it converges or diverges, which depends on the value of "r."
Series Ratio
The series ratio in a geometric series is the constant that determines how each subsequent term changes from the previous term. In our example, the series \(\sum_{n=0}^{\infty}(1.075)^{n}\) has a ratio \(r = 1.075\). This tells us that each term is 1.075 times the previous term.

### Understanding Convergence and Divergence
  • A geometric series converges if the absolute value of its ratio \( |r| \) is less than 1.
  • If \(|r| \geq 1\), the series diverges, like our example where \( |1.075| \geq 1 \).
Knowing how to determine convergence or divergence is crucial for determining the usefulness or applicability of a series in practical problems.
Symbolic Algebra Utility
Symbolic algebra utilities like Wolfram Alpha or other computer algebra systems (CAS) provide powerful tools for solving mathematical problems.
They can compute the convergence or divergence of series quickly, saving time and reducing human error that can occur in manual calculations.

### How to Use
  • Input the series expression into the utility, like \(\sum_{n=0}^{\infty}(1.075)^{n}\).
  • The utility processes the input and returns whether the series converges or diverges.
  • It's especially useful for complex series where manual checking is unwieldy.
These utilities can also offer visual graphing capabilities to illustrate series behavior more intuitively.
Mathematical Proof
Mathematical proof is a logical process that confirms the truth of a statement, ensuring it holds under all circumstances within its assumptions. Proving that a series converges or diverges is a critical application of mathematical proofs.

### Proving Series Divergence
  • For the geometric series \(\sum_{n=0}^{\infty}(1.075)^{n}\), we start by identifying the series ratio \( r = 1.075 \), which is crucial for determining behavior.
  • According to the convergence rule for geometric series, since \( |r| \geq 1 \), the series diverges.
This application of mathematical proof to confirm the series' divergence shows how proofs ensure precise and reliable results.

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Most popular questions from this chapter

Annuity A deposit of 100 dollars is made at the beginning of each month for 5 years in an account that pays \(10 \%\) interest, compounded monthly. Use a symbolic algebra utility to find the balance \(A\) in the account at the end of the 5 years. $$A=100\left(1+\frac{0.10}{12}\right)+\cdots+100\left(1+\frac{0.10}{12}\right)^{60}$$

Federal Debt It took more than 200 years for the United States to accumulate a 1 trillion dollars debt. Then it took just 8 years to get to 3 trillion dollars. The federal debt during the years 1990 through 2005 is approximated by the model \(a_{n}=0.003 n^{3}-0.07 n^{2}+0.63 n+3.08\) \(n=0,1,2,3, \ldots, 15 \)where \(a_{n}\) is the debt in trillions and \(n\) is the year, with \(n=0\) corresponding to \(1990 .\) (Source: U.S. Office of Management and Budget) (a) Write the terms of this finite sequence. (b) Construct a bar graph that represents the sequence.

Annuity A deposit of \(P\) dollars is made every month for \(t\) years in an account that pays an annual interest rate of \(r \%,\) compounded monthly. Let \(N=12 t\) be the total number of deposits. Show that the balance in the account after \(t\) years is $$A=P\left[\left(1+\frac{r}{12}\right)^{N}-1\right]\left(1+\frac{12}{r}\right), \quad t>0$$

Think About It Consider the sequence whose \(n\) th term \(a_{n}\) is given by $$a_{n}=\left(1+\frac{1}{n}\right)^{n}$$. Demonstrate that the terms of this sequence approach \(e\) by finding \(a_{1}, a_{10}, a_{100}, a_{1000},\) and \(a_{10,000}\).

Sales A company produces a new product for which it estimates the annual sales to be 8000 units. Suppose that in any given year \(10 \%\) of the units (regardless of age) will become inoperative. (a) How many units will be in use after \(n\) years? (b) Find the market stabilization level of the product.
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