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`sum_(n=3)^oo 1/(nlnn[ln(lnn)]^p)` Find the positive values of p for which the series converges.

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To find the positive values of p in which the series `sum_(n=3)^oo 1/(nln(n)(ln(ln(n)))^p)` , we may apply the integral test.

Integral test is applicable if f is positive, continuous, and decreasing function and `a_n=f(x)` . The infinite series `sum_(n=k)^oo a_n` converges if and only of the improper integral `int _k^oo f(x)dx ` converges to a finite number. If the integral diverges then the series also diverges.

For the infinite series `sum_(n=3)^oo 1/(nln(n)(ln(ln(x)))^p)` , we have:

`a_n=1/(nln(n)(ln(ln(n)))^p)`

Then,...

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