What should I put for using the integral test to determine whether the series is convergent or divergent?

DIVERGES and DIVERGENT doesn't work.

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Answer 1

The series:

#sum_(n=1)^oo a_n = sum_(n=1)^oo n/(n^2+5)#

is divergent.

Based on the integral test, the convergence of the series:

#sum_(n=1)^oo a_n = sum_(n=1)^oo n/(n^2+5)#

is equivalent to the convergence of the improper integral:

#int_1^oo x/(x^2+5)dx#
as the function #f(x) = x/(x^2+5)# in the interval #[1,+oo)# is positive, monotone decreasing and:
#f(n) = a_n#

Evaluate the indefinite integral:

#int x/(x^2+5) dx = 1/2int (d(x^2+5))/(x^2+5) =1/2ln(x^2+5) + C#

then:

#int_1^oo x/(x^2+5)dx = lim_(x->oo) 1/2ln(x^2+5) - 1/2ln6 = +oo#

So the series is divergent.

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Answer 2

When using the integral test to determine whether a series is convergent or divergent, you should follow these steps:

  1. Verify that the series is positive and decreasing for all terms beyond some point N.

  2. Integrate the function represented by the series from N to infinity.

  3. Determine whether the improper integral converges or diverges.

  4. If the integral converges, then the series converges. If the integral diverges, then the series also diverges.

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Answer from HIX Tutor

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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