How do you test the series #(lnk)/(k^2)# for convergence?

Answer 1

The series:

#sum_(k=1)^oo lnk/k^2#

is convergent.

To test the convergence of the series:

#sum_(k=1)^oo lnk/k^2#

we can use the integral test, with test function:

#f(x) = lnx/x^2#

Verify that the hypotheses of the integral test theorem are satisfied:

(i) #f(x)# is positive in #[1,+oo)#
(ii) #(df)/dx = (1-2lnx)/x^3# is negative in #[1,+oo)#

so the function is monotone decreasing in the interval

(iii) #lim_(x->oo) f(x) = 0#
(iv) #f(k) = lnk/k^2#

So the convergence of the series is equivalent to the convergence of the integral:

#int_1^oo lnx/x^2dx#

We start solving the indefinite integral by parts:

#int lnx/x^2dx = int lnx d(-1/x) = -lnx/x +int dx/x^2=-lnx/x-1/x+C#

so:

#int_1^oo lnx/x^2dx = [-lnx/x-1/x]_1^oo#
#int_1^oo lnx/x^2dx = 1 -lim_(x->oo) lnx/x - lim_(x->oo) 1/x =1#

The integral converges, so the series is also proven to be convergent.

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Answer 2

To test the series (\frac{\ln k}{k^2}) for convergence, you can use the integral test. Here's how:

  1. Integrate the function (f(x) = \frac{\ln x}{x^2}) from (1) to (n) to get (F(n)).
  2. If the integral (\int_1^\infty \frac{\ln x}{x^2} , dx) converges, then the series (\sum_{k=1}^\infty \frac{\ln k}{k^2}) converges. If the integral diverges, then the series diverges.

The integral test states that if (f(x)) is continuous, positive, and decreasing for (x \geq 1), then the convergence of the series (\sum_{k=1}^\infty f(k)) is equivalent to the convergence of the improper integral (\int_1^\infty f(x) , dx).

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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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