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Chapter 7: Q. 2TF (page 617)

Improper Integrals: Determine whether the following improper integrals converge or diverge.
$\int_{1}^{\infty} \frac{1}{x^{2}} d x .$

Short Answer

Expert verified

The series is a convergent series.

Step by step solution

Step 1. Given Information.

$G i v e n a n i n t e g r a l : \int_{1}^{\infty} \frac{1}{x^{2}} d x .$

Step 2. Solving the integral.

$\int_{1}^{\infty} \frac{1}{x^{2}} d x = {[\frac{- 1}{x}]}_{1}^{\infty} = [\frac{- 1}{\infty} - (\frac{- 1}{1})] = [0 + 1] = 1 . T h e r e f o r e, t h e s e r i e s i s a c o n v e r g e n t s e r i e s .$

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Most popular questions from this chapter

Prove that if $\sum_{k = 1}^{\infty} a_{k}$ converges to L and $\sum_{k = 1}^{\infty} b_{k}$ converges to M , then the series $\sum_{k = 1}^{\infty} (a_{k} + b_{k}) = L + M$ .

Prove Theorem 7.25. That is, show that the series $\sum_{k = 1}^{\infty} a_{k} a n d \sum_{k = M}^{\infty} a_{k}$ either both converge or both diverge. In addition, show that if $\sum_{k = M}^{\infty} a_{k}$ converges to L, then $\sum_{k = 1}^{\infty} a_{k}$ converges tolocalid="1652718360109" $a_{1} + a_{2} + a_{3} + . . . . + a_{M - 1} + L .$

If $\lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \infty$ and $\sum_{k = 1}^{\infty} a_{k}$ diverges, explain why we cannot draw any conclusions about the behavior of $\sum_{k = 1}^{\infty} b_{k}$ .

An Improper Integral and Infinite Series: Sketch the function $f (x) = \frac{1}{x}$ for x ≥ 1 together with the graph of the terms of the series $\sum_{k = 1}^{\infty} \frac{1}{k} .$ Argue that for every term $S_{n}$ of the sequence of partial sums for this series, $S_{n} > \int_{1}^{n + 1} \frac{1}{x} d x$ . What does this result tell you about the convergence of the series?

Find the values of x for which the series $\sum_{k = 0}^{\infty} x^{k}$ converges.

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Improper Integrals: Determine whether the following improper integrals converge or diverge.∫1∞1x2dx.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Solving the integral.

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Most popular questions from this chapter

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Improper Integrals: Determine whether the following improper integrals converge or diverge.
$\int_{1}^{\infty} \frac{1}{x^{2}} d x .$