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Chapter 9: Problem 6

Explain why the sequence of partial sums for a series with positive terms is an increasing sequence.

Short Answer

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Question: Demonstrate that the sequence of partial sums for a series with positive terms is an increasing sequence. Answer: Given a series with positive terms {a_n}, where a_n > 0 for all n, the sequence of partial sums {S_n} is calculated as S_n = a_1 + a_2 + ... + a_n. To prove that the sequence of partial sums is increasing, we must show that S_{n+1} > S_n for all n. Since a_{n+1} > 0, we can rewrite the inequality as S_{n+1} = S_n + a_{n+1} > S_n. Therefore, the sequence of partial sums {S_n} is an increasing sequence when dealing with a series with positive terms.

Step by step solution

01

Define a sequence with positive terms

Let's consider a sequence with positive terms: {a_n}, where a_n > 0 for all n (n is a positive integer).
02

Define partial sums for the sequence

Partial sums are the sums of the first n terms in the sequence. The sequence of partial sums is denoted by {S_n}, where S_n = a_1 + a_2 + ... + a_n for n = 1, 2, 3, ....
03

Show that the next partial sum is greater than the previous one

Let's consider two consecutive terms in the sequence of partial sums: S_n and S_{n+1}. We want to prove that S_{n+1} > S_n. We know that S_n = a_1 + a_2 + ... + a_n and S_{n+1} = a_1 + a_2 + ... + a_n + a_{n+1}. Since a_{n+1} > 0 (given that all terms are positive), we can rewrite the inequality as: S_{n+1} = S_n + a_{n+1} > S_n
04

Conclusion

Since S_{n+1} > S_n for all n, the sequence of partial sums {S_n} is an increasing sequence when dealing with a series with positive terms.

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Most popular questions from this chapter

Convergence parameter Find the values of the parameter \(p>0\) for which the following series converge. $$\sum_{k=2}^{\infty} \frac{1}{(\ln k)^{p}}$$

Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and/or graphing to make a conjecture about the value of the limit or determine that the limit does not exist. $$a_{n+1}=\sqrt{2+a_{n}} ; a_{0}=1, n=0,1,2, \dots$$

Consider the sequence \(\left\\{x_{n}\right\\}\) defined for \(n=1,2,3, \ldots\) by $$x_{n}=\sum_{k=n+1}^{2 n} \frac{1}{k}=\frac{1}{n+1}+\frac{1}{n+2}+\dots+\frac{1}{2 n}$$ a. Write out the terms \(x_{1}, x_{2}, x_{3}\) b. Show that \(\frac{1}{2} \leq x_{n}<1,\) for \(n=1,2,3, \ldots\) c. Show that \(x_{n}\) is the right Riemann sum for \(\int_{1}^{2} \frac{d x}{x}\) using \(n\) subintervals. d. Conclude that \(\lim _{n \rightarrow \infty} x_{n}=\ln 2\)

The expression where the process continues indefinitely, is called a continued fraction. a. Show that this expression can be built in steps using the recurrence relation \(a_{0}=1, a_{n+1}=1+1 / a_{n},\) for \(n=0,1,2,3, \ldots . .\) Explain why the value of the expression can be interpreted as \(\lim a_{n}\). b. Evaluate the first five terms of the sequence \(\left\\{a_{n}\right\\}\). c. Using computation and/or graphing, estimate the limit of the sequence. d. Assuming the limit exists, use the method of Example 5 to determine the limit exactly. Compare your estimate with \((1+\sqrt{5}) / 2,\) a number known as the golden mean. e. Assuming the limit exists, use the same ideas to determine the value of where \(a\) and \(b\) are positive real numbers.

For a positive real number \(p,\) how do you interpret \(p^{p^{p \cdot *}},\) where the tower of exponents continues indefinitely? As it stands, the expression is ambiguous. The tower could be built from the top or from the bottom; that is, it could be evaluated by the recurrence relations \(a_{n+1}=p^{a_{n}}\) (building from the bottom) or \(a_{n+1}=a_{n}^{p}(\text { building from the top })\) where \(a_{0}=p\) in either case. The two recurrence relations have very different behaviors that depend on the value of \(p\). a. Use computations with various values of \(p > 0\) to find the values of \(p\) such that the sequence defined by (2) has a limit. Estimate the maximum value of \(p\) for which the sequence has a limit. b. Show that the sequence defined by (1) has a limit for certain values of \(p\). Make a table showing the approximate value of the tower for various values of \(p .\) Estimate the maximum value of \(p\) for which the sequence has a limit.
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