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Questions and Answers
What is the primary characteristic that defines an alternating series?
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?
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?
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?
What does absolute convergence imply about the convergence of a series?
How does rearranging the terms of an absolutely convergent series affect its sum?
How does rearranging the terms of an absolutely convergent series affect its sum?
For what values of x does the power series $\sum_{n=0}^{\infty} x^n$ converge absolutely?
For what values of x does the power series $\sum_{n=0}^{\infty} x^n$ converge absolutely?
What distinguishes a conditionally convergent series from an absolutely convergent series?
What distinguishes a conditionally convergent series from an absolutely convergent series?
What is the effect of rearranging terms in a conditionally convergent series?
What is the effect of rearranging terms in a conditionally convergent series?
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?
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?
Given a power series $\sum c_n(x-a)^n$, what does the 'radius of convergence' indicate?
Given a power series $\sum c_n(x-a)^n$, what does the 'radius of convergence' indicate?
Flashcards
Alternating Series
Alternating Series
A series where the terms alternate in sign.
Absolute Convergence
Absolute Convergence
A series converges absolutely if the sum of the absolute values of its terms converges.
Convergence Condition
Convergence Condition
Series converges if |x| < 1 and diverges if |x| > 1.
Rearrangement of Series
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
Conditional Convergence
A series is conditionally convergent if it converges, but does not converge absolutely.
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