# How do you use the Integral test on the infinite series #sum_(n=1)^oo1/n^5# ?

By Integral Test,

Let us look at some details.

Let us evaluate the corresponding improper integral.

Since the integral

also converges by Integral Test.

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To use the Integral Test on the infinite series ( \sum_{n=1}^\infty \frac{1}{n^5} ), you compare it to the integral of the corresponding continuous function. The integral test states that if the integral of the function ( f(x) ) over the interval [1, ∞) converges or diverges, then the series ( \sum_{n=1}^\infty f(n) ) converges or diverges accordingly.

For this series, the corresponding function is ( f(x) = \frac{1}{x^5} ). Now, integrate ( f(x) ) over the interval [1, ∞):

[ \int_{1}^{\infty} \frac{1}{x^5} dx = \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^5} dx ]

[ = \lim_{b \to \infty} \left[ -\frac{1}{4x^4} \right]_{1}^{b} ]

[ = \lim_{b \to \infty} \left( -\frac{1}{4b^4} + \frac{1}{4} \right) ]

Since the limit converges to a finite value (1/4), the integral converges. Therefore, by the Integral Test, the series ( \sum_{n=1}^\infty \frac{1}{n^5} ) also converges.

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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.

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- How do you determine if the series the converges conditionally, absolutely or diverges given #Sigma ((-1)^(n+1))/(n^1.5)# from #[1,oo)#?
- How do you use the Root Test on the series #sum_(n=1)^oo((n^2+1)/(2n^2+1))^(n)# ?
- How do you test the series #Sigma 1/(nlnn)# from n is #[2,oo)# for convergence?

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