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QuestionMathematics

The Alternating Series Test has three conditions, all of which must be met, in order to guarantee that a series is convergent. Those three conditions are: Condition (1): The series is an alternating series (it has the form $\sum_{n= 1}^{\infty}(- 1)^{n+ 1} b_{n}$ where $b_{n}>0$ ); Condition (2): The absolute value of each term is less than or equal to that of the preceding term (the statement $b_{n+ 1} \leq b_{n}$ is true for all $n$ ); and Condition (3): The terms shrink toward zero for large $n$, or $\lim _{n \rightarrow \infty} b_{n}= 0$. Can the Alternating Series Test be applied to the series $\sum_{n= 1}^{\infty}(- 1)^{n+ 1} \frac{5 n}{4 n- 2}$ to show that it converges? Answer this question by picking the most appropriate answer choice from the list below. (3) No. Condition (1) is not satisfied (expanding the general term does not yield an alternating series), so this test (by itself) is inconclusive and a different convergence test must be used. (4) Yes. All three conditions are met, so the given series converges and no additional testing is required. (5) No. Condition (2) is not satisfied (the absolute value of each term is not shrinking relative to its predecessor), so this test (by itself) is inconclusive and a different convergence test must be used. (6) No. Condition (3) is not satisfied (the terms do not shrink toward zero for large $\boldsymbol{\tau}$ ), so this test (by itself) is inconclusive and a different convergence test must be used.

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Answer

Full Solution Locked

Step 1:
: Examine Condition 1 - Alternating Series

In this case, $(-1)^{n+1}$ ensures that the sign alternates, so Condition 1 is satisfied.

Step 2:
: Examine Condition 2 - Absolute Value of Each Term

Since this inequality does not hold for all $n$, Condition 2 is not satisfied.

We need to verify that the absolute value of each term is less than or equal to that of the preceding term. Let's calculate the absolute values of two consecutive terms: Now, compare the two expressions:

Step 3:
: Examine Condition 3 - Terms Shrink Toward Zero

\lim _{n \rightarrow \infty} \frac{5 n}{4 n-2}=\lim _{n \rightarrow \infty} \frac{5}{4}=\frac{5}{4}

Since the limit is not zero, Condition 3 is not satisfied.

Final Answer

No, the Alternating Series Test cannot be applied to the series $\sum_{n= 1}^{\infty}(- 1)^{n+ 1} \frac{5 n}{4 n- 2}$ to show that it converges, because Condition 2 and Condition 3 are not satisfied. A different convergence test must be used.

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Q
QuestionMathematics

Answer

Full Solution Locked

Step 1:
: Examine Condition 1 - Alternating Series

Step 2:
: Examine Condition 2 - Absolute Value of Each Term

Step 3:
: Examine Condition 3 - Terms Shrink Toward Zero

Final Answer

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QQuestionMathematics

AnswerHelpful

Full Solution Locked

Step 1:: Examine Condition 1 - Alternating Series

Step 2:: Examine Condition 2 - Absolute Value of Each Term

Step 3:: Examine Condition 3 - Terms Shrink Toward Zero

Final Answer

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Q
QuestionMathematics

Answer

Step 1:
: Examine Condition 1 - Alternating Series

Step 2:
: Examine Condition 2 - Absolute Value of Each Term

Step 3:
: Examine Condition 3 - Terms Shrink Toward Zero