Alternating Series Convergence

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Questions and Answers

What is the primary characteristic that defines an alternating series?

  • The series converges absolutely.
  • The series has a common ratio between consecutive terms.
  • The series consists of only positive terms.
  • The series consists of terms that alternate in sign. (correct)

Which of the following conditions is sufficient to prove that an alternating series converges, according to the alternating series test?

  • The terms of the series are bounded.
  • The terms of the series increase in magnitude.
  • The terms of the series alternate in sign and approach a non-zero constant.
  • The terms of the series decrease in magnitude and approach zero. (correct)

If an alternating series converges conditionally, what can be said about its positive and negative terms?

  • The series of positive terms converges, and the series of negative terms diverges.
  • Both the series of positive terms and the series of negative terms converge.
  • The series of positive terms diverges, and the series of negative terms converges.
  • Both the series of positive terms and the series of negative terms diverge. (correct)

What does absolute convergence imply about the convergence of a series?

<p>It implies that the series converges regardless of the signs of its terms. (C)</p> Signup and view all the answers

How does rearranging the terms of an absolutely convergent series affect its sum?

<p>The sum remains the same. (C)</p> Signup and view all the answers

For what values of x does the power series $\sum_{n=0}^{\infty} x^n$ converge absolutely?

<p>$-1 &lt; x &lt; 1$. (C)</p> Signup and view all the answers

What distinguishes a conditionally convergent series from an absolutely convergent series?

<p>A conditionally convergent series converges only when the alternating signs are considered, but diverges when absolute values are taken. (A)</p> Signup and view all the answers

What is the effect of rearranging terms in a conditionally convergent series?

<p>It may converge to a different sum or diverge. (D)</p> Signup and view all the answers

Consider an alternating series where the absolute value of the terms, $|a_n|$, is non-increasing and approaches zero. Which of the following must be true?

<p>The series converges, but not necessarily absolutely or conditionally. (B)</p> Signup and view all the answers

Given a power series $\sum c_n(x-a)^n$, what does the 'radius of convergence' indicate?

<p>The largest distance from <code>a</code> for which the series converges. (D)</p> Signup and view all the answers

Flashcards

Alternating Series

A series where the terms alternate in sign.

Absolute Convergence

A series converges absolutely if the sum of the absolute values of its terms converges.

Convergence Condition

Series converges if |x| < 1 and diverges if |x| > 1.

Rearrangement of Series

A rearrangement of a series is a series with the same terms, but in a different order.

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Conditional Convergence

A series is conditionally convergent if it converges, but does not converge absolutely.

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