Using the integral test, how do you show whether #sum 3/(n sqrt(ln(n)))# diverges or converges from n=1 to infinity?
See the explanation.
so
The integral and the series diverge.
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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:
- Formulate the corresponding improper integral.
- 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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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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