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Chapter 7: Q. 93 (page 593)

Prove that if ${\{a_{k}\}}_{k = 1}^{\infty}$ is a sequence of positive real numbers, then the sequence ${\{S_{n}\}}_{n = 1}^{\infty}$ , where the sequence Sn = a1 + a2 +···+an, is an increasing sequence.

Short Answer

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Proved.

Step by step solution

01

Step 1. Given

Consider the sequences ${\{a_{k}\}}_{k = 1}^{\infty}$ .

02

Step 2. Proof

$The sequenc e S_{n} = a_{1} + a_{2} + a_{3} + . . . + a_{n} . . . . . . (1) Change n to n + 1 in equation (1) S_{n + 1} = a_{1} + a_{2} + a_{3} + . . . + a_{n} + a_{n + 1} . . . . . (2) Subtract (1) from (2) S_{n + 1} - S_{n} = a_{1} + a_{2} + a_{3} + . . . + a_{n} + a_{n + 1} - (a_{1} + a_{2} + a_{3} + . . . + a_{n}) = a . Hence S_{n + 1} - S_{n} = a_{n + 1} Since S_{n + 1} \geq S_{n} Hence the sequence is increasing sequence .$

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Most popular questions from this chapter

True/False:

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: If $a_{k} \to 0$ , then $\sum_{k = 1}^{\infty} a_{k}$ converges.

(b) True or False: If $\sum_{k = 1}^{\infty} a_{k}$ converges, then $a_{k} \to 0$ .

(c) True or False: The improper integral $\int_{1}^{\infty} f (x) d x$ converges if and only if the series $\sum_{k = 1}^{\infty} f (k)$ converges.

(d) True or False: The harmonic series converges.

(e) True or False: If $p > 1$ , the series $\sum_{k = 1}^{\infty} k^{- p}$ converges.

(f) True or False: If $f (x) \to 0$ as $x \to \infty$ , then $\sum_{k = 1}^{\infty} f (k)$ converges.

(g) True or False: If $\sum_{k = 1}^{\infty} f (k)$ converges, then $f (x) \to 0$ as $x \to \infty$ .

(h) True or False: If $\sum_{k = 1}^{\infty} a_{k} = L$ and ${S_{n}}$ is the sequence of partial sums for the series, then the sequence of remainders ${L - S_{n}}$ converges to $0$ .

Let f(x) be a function that is continuous, positive, and decreasing on the interval $[1, \infty)$ such that role="math" localid="1649081384626" $\underset{x \to \infty}{l i m} f (x) = α > 0$ . What can the divergence test tell us about the series $\sum_{k = 1}^{\infty} f (k)$ ?

Which p-series converge and which diverge?

Prove Theorem 7.31. That is, show that if a function a is continuous, positive, and decreasing, and if the improper integral $\int_{1}^{\infty} a (x) d x$ converges, then the nth remainder, $R_{n}$ , for the series $\sum_{k = 1}^{\infty} a (k)$ is bounded by $0 \leq R_{n} = \sum_{k = n + 1}^{\infty} a (k) \leq \int_{n}^{\infty} a (x) d x$

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A divergent series $\sum_{k = 1}^{\infty} a_{k}$ in which $a_{k} \to 0$ .

(b) A divergent p-series.

(c) A convergent p-series.

See all solutions

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