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Chapter 8: Problem 75

Find the values of the parameter \(p>0\) for which the following series converge. $$\sum_{k=1}^{\infty} \frac{k p^{k}}{k+1}$$

Short Answer

Expert verified
Based on the Ratio Test and our calculations, the series \(\sum_{k=1}^{\infty} \frac{k p^{k}}{k+1}\) converges for the values of the positive parameter \(p\) in the range of \(0<p<1\).

Step by step solution

01

Apply the Ratio Test

The Ratio Test states that if the limit $$\lim_{k\to\infty} \left|\frac{a_{k+1}}{a_k}\right| < 1$$ exists and is less than 1, then the series \(\sum_{k=1}^{\infty} a_k\) converges. If the limit is greater than 1, then the series diverges. In our case, we have \(a_k = \frac{k p^{k}}{k+1}\), so $$\frac{a_{k+1}}{a_k} = \frac{\frac{(k+1)p^{k+1}}{k+2}}{\frac{kp^k}{k+1}} = \frac{(k+1)p}{k+2}.$$ We need the limit of this expression when \(k\to\infty\): $$\lim_{k\to\infty} \frac{(k+1)p}{k+2} = \lim_{k\to\infty} \frac{p}{1+\frac{1}{k}} = p.$$ Now, if we find a suitable condition for \(p\) that makes this limit to be less than 1, we can conclude the series converges for those values of \(p\).
02

Determine the condition for convergence

From the Ratio Test analysis in Step 1, we found that: $$\lim_{k\to\infty} \frac{(k+1)p}{k+2} = p.$$ Now, we compare this limit with 1: - If p < 1, the series converges. - If p > 1, the series diverges. With this information, we can answer the problem by specifying the values of the parameter \(p>0\) for which the series converges.
03

State the conclusion

From the analysis conducted in the previous steps, we can determine that the given series converges for the values of \(p\) such that \(0<p<1\).

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Most popular questions from this chapter

Marie takes out a \(\$ 20,000\) loan for a new car. The loan has an annual interest rate of \(6 \%\) or, equivalently, a monthly interest rate of \(0.5 \% .\) Each month, the bank adds interest to the loan balance (the interest is always \(0.5 \%\) of the current balance), and then Marie makes a \(\$ 200\) payment to reduce the loan balance. Let \(B_{n}\) be the loan balance immediately after the \(n\) th payment, where \(B_{0}=\$ 20,000\). a. Write the first five terms of the sequence \(\left\\{B_{n}\right\\}\). b. Find a recurrence relation that generates the sequence \(\left\\{B_{n}\right\\}\). c. Determine how many months are needed to reduce the loan balance to zero.

The sequence \(\\{n !\\}\) ultimately grows faster than the sequence \(\left\\{b^{n}\right\\},\) for any \(b>1,\) as \(n \rightarrow \infty .\) However, \(b^{n}\) is generally greater than \(n !\) for small values of \(n\). Use a calculator to determine the smallest value of \(n\) such that \(n !>b^{n}\) for each of the cases \(b=2, b=e,\) and \(b=10\).

Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and/or graphing to make a conjecture about the limit of the sequence or state that the sequence diverges. $$a_{n+1}=\frac{1}{2} a_{n}+2 ; a_{0}=5$$

Express each sequence \(\left\\{a_{n}\right\\}_{n=1}^{\infty}\) as an equivalent sequence of the form \(\left\\{b_{n}\right\\}_{n=3}^{\infty}\). $$\\{2 n+1\\}_{n=1}^{\infty}$$

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