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Chapter 7: Problem 112

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.

Short Answer

Expert verified
The statement is false. The series \(\sum_{n=1}^{\infty} \frac{n}{1000(n+1)}\) converges.

Step by step solution

01

Write down the series

The series given is \(\sum_{n=1}^{\infty} \frac{n}{1000(n+1)}\)
02

Simplify the series

The series can be simplified to \(\sum_{n=1}^{\infty} \frac{1}{1000} - \frac{1}{1000(n+1)}\)
03

Apply the Divergence Test

We find the limit of \( \frac{1}{1000} - \frac{1}{1000(n+1)} \) as \(n\) approaches infinity, which is \( \lim_{n->\infty} \left( \frac{1}{1000} - \frac{1}{1000(n+1)} \right) = 0\). Since the limit is equal to zero, the Divergence Test gives us no information.
04

Apply the test for Telescoping Series

Notice that the series can be written as a difference of fractions, which suggests that it can be a telescoping series. In a telescoping series the sum of the terms simplify significantly and in this case, it simplifies to \( \frac{1}{1000}-0 = \frac{1}{1000} \) which is finite, hence the series converges.

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Most popular questions from this chapter

(a) You delete a finite number of terms from a divergent series. Will the new series still diverge? Explain your reasoning. (b) You add a finite number of terms to a convergent series. Will the new series still converge? Explain your reasoning.

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. \(0.75=0.749999 \ldots \ldots\)

Find the sum of the convergent series. $$ \sum_{n=0}^{\infty}\left(-\frac{1}{2}\right)^{n} $$

Find the values of \(x\) for which the infinite series \(1+2 x+x^{2}+2 x^{3}+x^{4}+2 x^{5}+x^{6}+\cdots\) converges. What is the sum when the series converges?

Consider the formula \(\frac{1}{x-1}=1+x+x^{2}+x^{3}+\cdots\) Given \(x=-1\) and \(x=2\), can you conclude that either of the following statements is true? Explain your reasoning. (a) \(\frac{1}{2}=1-1+1-1+\cdots\) (b) \(-1=1+2+4+8+\cdots\)
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