# How do you determine if the series the converges conditionally, absolutely or diverges given #Sigma ((-1)^(n+1))/(n^1.5)# from #[1,oo)#?

The series:

#sum_(n=1)^oo (-1)^(n+1)/n^(3/2)#

is absolutely convergent

Given the series:

we can test for convergence the series:

If this series converges, then the series (1) converges absolutely ( and also conditionally ).

We can apply the integral test to the series (2) using the function:

Se we calculate:

which means that the series (1) is absolutely convergent.

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To determine the convergence behavior of the series ∑((-1)^(n+1))/(n^1.5) from n = 1 to infinity:

- First, check the absolute convergence by considering the series formed by taking the absolute value of each term: ∑(1/n^1.5).
- Use the p-series test, which states that ∑(1/n^p) converges if p > 1 and diverges if p ≤ 1.
- Since p = 1.5, which is greater than 1, the series ∑(1/n^1.5) converges.
- If the absolute series converges, but the original series does not, then the original series converges conditionally.
- If both the absolute series and the original series converge, the original series converges absolutely.
- If neither the absolute series nor the original series converges, the original series diverges.

In this case, since ∑(1/n^1.5) converges, the given series ∑((-1)^(n+1))/(n^1.5) converges absolutely.

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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.

- #lim_(n->oo)(1/(1 xx 2)+1/(2 xx 3)+1/(3 xx4) + cdots + 1/(n(n+1)))#?
- How do you test the series #Sigma n/((n+1)(n^2+1))# from n is #[0,oo)# for convergence?
- How do you test the alternating series #Sigma (-1)^n(sqrt(n+1)-sqrtn)# from n is #[1,oo)# for convergence?
- How do you use the comparison test for #sum (((ln n)^3) / (n^2))# n=1 to #n=oo#?
- How do you test for convergence of #Sigma n e^-n# from #n=[1,oo)#?

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