How do you test the alternating series #Sigma (1)^(n+1)(1+1/n)# from n is #[1,oo)# for convergence?
The alternating series test basically says this:
If a series is alternating, then as long as the ABSOLUTE VALUE of each term is less than the ABSOLUTE VALUE of the previous term, the series will converge.
Therefore, the series diverges.
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To test the alternating series ( \sum_{n=1}^{\infty} (1)^{n+1} \left(1 + \frac{1}{n}\right) ) for convergence, you can use the Alternating Series Test. This test states that if the series alternates in sign and the absolute value of its terms decreases monotonically to zero, then the series converges.

Check the alternating sign: The series alternates in sign because it has the term ( (1)^{n+1} ), which changes sign as ( n ) increases.

Check the monotonic decrease: Consider the absolute value of the terms of the series, which is ( \left1 + \frac{1}{n}\right = 1 + \frac{1}{n} ). As ( n ) increases, ( \frac{1}{n} ) decreases, so ( 1 + \frac{1}{n} ) also decreases monotonically to ( 1 ).
Since the series alternates in sign and the absolute value of its terms decreases monotonically to zero, the Alternating Series Test guarantees convergence. Therefore, the series ( \sum_{n=1}^{\infty} (1)^{n+1} \left(1 + \frac{1}{n}\right) ) converges.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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