Using the integral test, how do you show whether #sum 3/(n sqrt(ln(n)))# diverges or converges from n=1 to infinity?

Answer 1

See the explanation.

Let #f(x) = 3/(xsqrtln(x))#. #" "#(So that #f(n) = a_n#.)
It is clear the #f# is continuous and decreasing on #[1,oo)#

so

#sum 3/(n sqrt(ln(n)))# converges or diverges
along with #int_1^oo 3/(xsqrtln(x)) dx#
#int_1^oo 3/(xsqrtln(x)) dx = lim_(brarroo)int_1^b 3/(xsqrtln(x)) dx #
# = 3lim_(brarroo)int_1^b (ln(x))^(-1/2) 1/x dx #
# = 3lim_(brarroo) 2(ln(x))^(1/2)]_1^b #
# = 6lim_(brarroo) (sqrtln(b) - sqrtln(1)) #
# = 6lim_(brarroo) sqrtln(b) = oo#

The integral and the series diverge.

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Answer 2

To determine whether the series ( \sum_{n=1}^{\infty} \frac{3}{n \sqrt{\ln(n)}} ) converges or diverges using the integral test, we follow these steps:

  1. Formulate the corresponding improper integral.
  2. Determine whether the integral converges or diverges.

Let's proceed with step 1:

The corresponding integral for the series is given by: [ \int_{1}^{\infty} \frac{3}{x \sqrt{\ln(x)}} , dx ]

Now, let's evaluate this integral to determine convergence or divergence.To evaluate the integral ( \int_{1}^{\infty} \frac{3}{x \sqrt{\ln(x)}} , dx ), we can use the substitution method. Let ( u = \ln(x) ), then ( du = \frac{1}{x} , dx ).

Making the substitution, we have: [ \int_{1}^{\infty} \frac{3}{x \sqrt{\ln(x)}} , dx = \int_{0}^{\infty} \frac{3}{\sqrt{u}} , du ]

Now, integrating ( \frac{3}{\sqrt{u}} ) with respect to ( u ) yields: [ \int_{0}^{\infty} \frac{3}{\sqrt{u}} , du = \lim_{b \to \infty} [2 \sqrt{u}]{0}^{b} = \lim{b \to \infty} [2 \sqrt{b} - 2 \sqrt{0}] ] [ = 2 \lim_{b \to \infty} \sqrt{b} = \infty ]

Since the improper integral diverges, by the integral test, the series ( \sum_{n=1}^{\infty} \frac{3}{n \sqrt{\ln(n)}} ) 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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