How do you determine if the series the converges conditionally, absolutely or diverges given #Sigma ((1)^(n))/(sqrt(n+4))# from #[1,oo)#?
The series:
is convergent but not absolutely convergent.
This is an alternating series in the form:
thus based on Leibniz's theorem, it is convergent if:
In our case:
and
so both conditions are satisfied and the series is convergent.
Consider now the series of absolute values:
Using the limit comparison test we can see that as:
is convergent but not absolutely convergent.
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To determine the convergence of the series ( \sum_{n=1}^{\infty} \frac{(1)^n}{\sqrt{n+4}} ) from ( n = 1 ) to infinity:

Absolute Convergence: Test if the absolute value of the series converges. If ( \sum_{n=1}^{\infty} \left \frac{(1)^n}{\sqrt{n+4}} \right ) converges, the series is absolutely convergent.

Conditional Convergence: If the series converges but the absolute value of the series diverges, then the series is conditionally convergent.

Divergence: If neither the original series nor its absolute value converges, the series diverges.
To determine absolute convergence, consider ( \sum_{n=1}^{\infty} \left \frac{(1)^n}{\sqrt{n+4}} \right ) and evaluate whether it converges or diverges. If it converges, the original series converges absolutely. If it diverges, further analysis is needed to determine conditional convergence or divergence.
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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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