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How do you test the series #(lnk)/(k^2)# for convergence?
1 Answer
Explanation:
To test the convergence of the series:
#sum_(k=1)^oo lnk/k^2#
we can use the integral test, with test function:
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#f(x) = lnx/x^2#
Verify that the hypotheses of the integral test theorem are satisfied:
(i) #f(x)# is positive in #[1,+oo)#
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(ii) #(df)/dx = (1-2lnx)/x^3# is negative in #[1,+oo)#
so the function is monotone decreasing in the interval
Infographics for pages 3 1 – visualization graphics for pages. (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# Network utility x 4 2 1.
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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