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Chapter 9: Problem 73

State the definitions of convergent and divergent series.

Short Answer

Expert verified
A convergent series is a series whose partial sums tend to a finite limiting value as the number of terms approaches infinity. On the other hand, a divergent series is a series whose partial sums do not tend to a limit as the number of terms approaches infinity.

Step by step solution

01

Definition of Convergent Series

A series \(\sum_{n=1}^\infty a_n\) is said to be convergent if the sequence of its partial sums \(S_n = \sum_{i=1}^n a_i\) tends to a limiting value 'L' as \(n \rightarrow \infty\). In other words, \(\lim_{n \rightarrow \infty} S_n = L\). If such a limit 'L' exists and is finite, the series is convergent and 'L' is called the sum of the series.
02

Definition of Divergent Series

A series \(\sum_{n=1}^\infty a_n\) is said to be divergent if the sequence of its partial sums \(S_n = \sum_{i=1}^n a_i\) does not tend to a limit as \(n \rightarrow \infty\). That is either \(\lim_{n \rightarrow \infty} S_n = \pm \infty\) (the series is said to be divergent to \(\pm \infty\)) or the limit does not exist in any sense.

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Key Concepts

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

Sequence of Partial Sums
To understand the concept of a convergent or divergent series, we begin with the sequence of partial sums. Imagine we have a series, which is just a fancy term for the sum of a sequence of numbers (think of a list like 1, 1/2, 1/4, and so on). As we keep adding more and more terms, we start creating what's called a sequence of partial sums. Now, these aren't just random; they're the key to unlocking the behavior of the series.

Here's a simpler way to picture it: if you saved money in a piggy bank over time, each deposit would represent a term in your series, and the total amount in the piggy bank at any moment would be a partial sum. The pattern your savings follow as you keep adding money could tell you if you're going to be a millionaire (convergent) or if you'll keep adding money forever without hitting a cap (divergent).
Limit of a Series
Now that we've talked about sequences of partial sums, we can tackle the limit of a series. This is where we get down to nitty-gritty details. If the sequence of partial sums steadies out and approaches a specific value as we add more terms infinitely, we declare that the series has a limit. Think of it like approaching the finish line in a race. If you can imagine reaching it, even if you're just taking tiny steps forward, that finish line is the limit. But if there's no finish line in sight, and you're just running forever toward an endless horizon, that signifies a series without a limit.

In mathematical terms, if the total amount in our piggy bank example (our partial sums) gets closer and closer to a number (let's say $1000) the closer we look, that's our limit. If it turns out you could count forever and never reach a stopping point, then the series diverges, and there's no limit to talk about.
Sum of the Series
When we mention the sum of the series, you might think it's as easy as just adding up all the terms. That's true for a finite series, but for an infinite series, it's a bit more complex. In this context, the sum of the series is the value that the sequence of partial sums is approaching. It's the final amount of money you can expect to have in your piggy bank if you kept saving indefinitely.

If the sequence has a nice, well-behaved pattern, and there's a clear value it's getting closer to, then we say the series converges to that sum. Imagine pouring water into a glass; if the water level rises to the brim and stops, the height of the water is like the sum of the series. But if that glass never fills up, because it's bottomless or the water level just keeps bobbing up and down, then we don't have a sum to speak of. That's when we say the series diverges, and there's no single answer to 'How much water (sum) is in the glass?'

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Most popular questions from this chapter

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The series \(\sum_{n=1}^{\infty} \frac{n}{1000(n+1)}\) diverges.

Fibonacci Sequence In a study of the progeny of rabbits, Fibonacci (ca. \(1170-\) ca. 1240 ) encountered the sequence now bearing his name. It is defined recursively by \(a_{n+2}=a_{n}+a_{n+1}, \quad\) where \(\quad a_{1}=1\) and \(a_{2}=1\). (a) Write the first 12 terms of the sequence. (b) Write the first 10 terms of the sequence defined by $$ b_{n}=\frac{a_{n+1}}{a_{n}}, \quad n \geq 1 . $$ (c) Using the definition in part (b), show that $$ b_{n}=1+\frac{1}{b_{n-1}} $$ (d) The golden ratio \(\rho\) can be defined by \(\lim _{n \rightarrow \infty} b_{n}=\rho\). Show that \(\rho=1+1 / \rho\) and solve this equation for \(\rho\).

determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the interval of convergence for \(\sum_{n=0}^{\infty} a_{n} x^{n}\) is \((-1,1)\), then the interval of convergence for \(\sum_{n}^{\infty} a_{n}(x-1)^{n}\) is \((0,2)\).

Prove that if \(\left\\{s_{n}\right\\}\) converges to \(L\) and \(L>0\), then there exists ? number \(N\) such that \(s_{n}>0\) for \(n>N\).

Marketing An electronic games manufacturer producing a new product estimates the annual sales to be 8000 units. Each year \(10 \%\) of the units that have been sold will become inoperative. So, 8000 units will be in use after 1 year, \([8000+0.9(8000)]\) units will be in use after 2 years, and so on. How many units will be in use after \(n\) years?
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