How do you test the alternating series #Sigma (-1)^(n+1)n/(10n+5)# from n is #[1,oo)# for convergence?
Diverges by the Divergence Test.
However, take the limit of the overall sequence:
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To test the alternating series ( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}n}{10n+5} ) for convergence, you can use the Alternating Series Test.
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Check the terms ( \frac{n}{10n+5} ) to see if they are:
- Positive
- Decreasing (strictly decreasing)
- Tending to zero as ( n ) approaches infinity.
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Verify the alternating condition:
- ( (-1)^{n+1} ) alternates sign.
If both conditions are met, then the series converges.
For the given series:
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The terms ( \frac{n}{10n+5} ) are positive, decreasing, and tend to zero as ( n ) approaches infinity because as ( n ) increases, the ( 10n ) term dominates the ( 5 ) term in the denominator, making the fraction tend to ( \frac{n}{10n} = \frac{1}{10} ), which tends to zero.
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The series alternates sign with ( (-1)^{n+1} ).
Therefore, the alternating series ( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}n}{10n+5} ) converges by the Alternating Series Test.
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When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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- How do you Find the sum of the harmonic series?
- Using the integral test, how do you show whether #sum1/[(n^2)+4)# diverges or converges from n=1 to infinity?
- What do you do if the Alternating Series Test fails?

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