   Chapter 11.7, Problem 19E

Chapter
Section
Textbook Problem

Test the series for convergence or divergence.19. ∑ n = 1 ∞ ( − 1 ) n ln   n n

To determine

To test: Whether the series convergence or divergence.

Explanation

Result used:

(1)“If the alternating series n=1(1)n1bn=b1b2+b3b4+...   bn>0 satisfies the conditions bn+1bn   for all n and limnbn=0 , then the series is convergent; otherwise, the series is divergent.”

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

Calculation:

Consider the function f(x)=lnxx=(x12lnx) from the series n=1an=n=1lnnn .

Obtain the derivative of the function.

f(x)=x12ddx(lnx)+(lnx)ddx(x12)    [ddx[f(x)g(x)]=dfdxg(x)+f(x)dgdx]=1x1x+(

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