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.

Check the terms ( \frac{n}{10n+5} ) to see if they are:
 Positive
 Decreasing (strictly decreasing)
 Tending to zero as ( n ) approaches infinity.

Verify the alternating condition:
 ( (1)^{n+1} ) alternates sign.
If both conditions are met, then the series converges.
For the given series:

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.

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 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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