   Chapter 11.7, Problem 7E

Chapter
Section
Textbook Problem

# Test the series for convergence or divergence.7. ∑ n = 2 ∞ 1 n ln   n

To determine

To test: Whether the series convergence or divergence.

Explanation

Definition used:

The improper integral abf(x)dx is divergent if the limit does not exist.

Result used:

(1) If the function f(x) is continuous, positive and decreasing on [1,) and let an=f(n). then the series n=1an is divergent if and only if the improper integral 1f(x)dx is divergent.

(2) The function f(x) is decreasing function if f(x)<0.

Calculation:

Consider the function f(x)=1xlnx=(x1lnx12) from the series n=21nlnn.

Obtain the derivative of the function.

f(x)=x1ddx(lnx12)+(lnx12)ddx(x1)    [ddx[f(x)g(x)]=dfdxg(x)+f(x)dgdx]=x1(1x12)ddx(x12)+(lnx12)(x11)         [d[f(x)]ndx=n[f(x)]n1f(x)]=(12)(x12)x321x2lnx=1x2(12+1lnx)

Since f(x)<0 for n2 and by the Result (2), the given function is a decreasing function

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