   Chapter 11.7, Problem 36E

Chapter
Section
Textbook Problem

# Test the series for convergence or divergence.36. ∑ n = 2 ∞ 1 ( ln   n ) ln   n

To determine

To test: Whether the series is convergence or divergence.

Explanation

Result used:

(1) “Suppose that an and bn are the series with positive terms,

(a) If bn is convergent and anbn for all n, then an is also convergent.

(b) If bn is divergent and anbn for all n, then an is also divergent.”

(2) The p-series n=11np is convergent if p>1 and divergent if p1.

Calculation:

The given series n=1an=n=21(lnn)lnn.

(lnn)lnn=(elnlnn)lnn=(elnn)lnlnn=nlnlnn

Since lnlnn as n, lnlnn>2 for sufficiently large n. Choose the n as sufficiently large

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