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Chapter 7: Problem 124

Show that the Ratio Test and the Root Test are both inconclusive for the logarithmic \(p\) -series \(\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{p}}\)

Short Answer

Expert verified
Both the Ratio Test and the Root Test are inconclusive for the logarithmic \(p\)-series \(\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{p}}\) since they both result in a limit of 1.

Step by step solution

01

Applying the Ratio Test

For the Ratio Test, we take the limit as \(n\) goes to infinity of the absolute value of the ratio between the \((n+1)\)th term and the \(n\)th term of the series. For our series, this means we need to solve \(\lim_{{n \to \infty}} \left| \frac{n(\ln n)^{p}}{(n+1)(\ln(n+1))^{p}} \right|\). As you simplify, you'll see that the limit equals 1, which is inconclusive for the Ratio Test (we need the limit to be less than 1 for convergence and greater than 1 for divergence).
02

Applying the Root Test

For the Root Test, we calculate the limit as \(n\) tends to infinity of the \(n\)th root of the absolute value of the \(n\)th term of the series. For our given series, we compute \(\lim_{{n \to \infty}} \left( \frac{1} {n(\ln n)^{p}} \right)^{1/n}\). As you simplify, you'll end up with 1 which once again is inconclusive for the test. The Root Test only determines a series converges if the limit is less than 1, and diverges if it is greater than 1.
03

Conclusion

With both tests returning a limit of 1, neither test is able to decisively determine whether our series converges or diverges. Therefore, the Ratio Test and the Root Test are both inconclusive for our logarithmic \(p\)-series.

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Most popular questions from this chapter

The random variable \(\boldsymbol{n}\) represents the number of units of a product sold per day in a store. The probability distribution of \(n\) is given by \(P(n) .\) Find the probability that two units are sold in a given day \([P(2)]\) and show that \(P(1)+P(2)+P(3)+\cdots=1\). $$ P(n)=\frac{1}{3}\left(\frac{2}{3}\right)^{n} $$

Suppose that \(\sum a_{n}\) and \(\sum b_{n}\) are series with positive terms. Prove that if \(\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=\infty\) and \(\sum b_{n}\) diverges, \(\sum a_{n}\) also diverges.

Given two infinite series \(\sum a_{n}\) and \(\sum b_{n}\) such that \(\sum a_{n}\) converges and \(\sum b_{n}\) diverges, prove that \(\sum\left(a_{n}+b_{n}\right)\) diverges.

A fair coin is tossed repeatedly. The probability that the first head occurs on the \(n\) th toss is given by \(P(n)=\left(\frac{1}{2}\right)^{n},\) where \(n \geq 1\) (a) Show that \(\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^{n}=1\). (b) The expected number of tosses required until the first head occurs in the experiment is given by \(\sum_{n=1}^{\infty} n\left(\frac{1}{2}\right)^{n}\) Is this series geometric? (c) Use a computer algebra system to find the sum in part (b).

In Exercises \(53-68,\) determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{n+10}{10 n+1} $$
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