Consider the following series: ∑n=1∞(−1)n1n Use the Alternating Series Test to show this series converges.

Concept Used: Alternate Series Test: Suppose that we have a series ∑an and either an=(−1)nbn or an=(−1)n+1bn Where, bn≥0 for all n Then if, 1. limn→∞bn=0 2. {bn} is a decreasing sequence. The series ∑an is convergent. Explanation: Given series is: ∑n=1∞(−1)n1n Identify the bn for the test. bn=1n Now we will test the two conditions: limn→∞bn=limn→∞1n=0 bn=1n&gt;1n+1=bn+1 Both conditions are met. Series must be convergent.

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High School
Calculus and Analysis
Calculus 2
Series

Tolnaio

Answered question

2020-11-30

Consider the following series:

\sum_{n = 1}^{\infty} (- 1)^{n} \frac{1}{n}

Use the Alternating Series Test to show this series converges.

Answer & Explanation

diskusje5

Skilled2020-12-01Added 82 answers

Concept Used:
Alternate Series Test:
Suppose that we have a series

\sum a_{n}

and either

a_{n} = (- 1)^{n} b_{n}

a_{n} = (- 1)^{n + 1} b_{n}

Where,

b_{n} \geq 0

for all n
Then if,
1.

lim_{n \to \infty} b_{n} = 0

{b_{n}}

is a decreasing sequence.
The series

\sum a_{n}

is convergent.
Explanation:
Given series is:

\sum_{n = 1}^{\infty} (- 1)^{n} \frac{1}{n}

Identify the

b_{n}

for the test.

b_{n} = \frac{1}{n}

Now we will test the two conditions:

lim_{n \to \infty} b_{n} = lim_{n \to \infty} \frac{1}{n} = 0

b_{n} = \frac{1}{n} > \frac{1}{n + 1} = b_{n + 1}

Both conditions are met.
Series must be convergent.

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