Chapter 8: Infinite Sequences and Series
Section 8.3: Convergence Tests
Example 8.3.10
Determine if the series ∑n=2∞lnnn2 diverges, converges absolutely, or converges conditionally.
If it converges conditionally, determine if it also converge absolutely.
Solution
Mathematical Solution
The Integral test will establish the absolute convergence of the given series if the following three calculations are performed.
limn→∞lnnn2 = 0
ddxlnxx2=2 lnxx31−lnx
∫3∞lnxx2 ⅆx = 13ln32+23ln3+23
The first shows that an→0 as n→∞. The second shows that the function fx=lnn/n2 is monotone decreasing for lnx>1, or x>e≐2.7.
Together, the first two results show that the Integral test can be applied. The third then shows that by the Integral test, the given series converges absolutely.
Maple Solution
Figure 8.3.10(a) contains a graph of the function fx=lnx/x2 (in red) and of its derivative (in green).
On the basis of this graph, it may be conjectured that f is monotone decreasing and bounded below by zero, provided x≥3. (The derivative appears to be negative for x>3.)
Consequently, the Integral test may be tried as a test for absolute convergence, provided the integration starts from, say, x=3.
To this end:
Calculus palette: Definite integral template Context Panel: Evaluate and Display Inline
module() local F,p,N; F:=(ln(x)/x)^2; N:=10; p:=plot([F,diff(F,x)],x=1..N,color=[red,green],view=[0..N,default],tickmarks=[N,4],labels=[x,y]); print(p); end module:
Figure 8.3.10(a) Graph of fx (red) and f′x (green)
Since the integral converges, the series converges absolutely.
The following two calculations support the use of the Integral test.
Calculus palette: Limit operator
Context Panel: Evaluate and Display Inline
Calculus palette: Differentiation operator
ⅆⅆ x lnxx2 = 2lnxx3−2lnx2x3
A modicum of algebra puts the derivative into the form 2 lnxx31−lnx, from which it can be seen that the derivative is negative for 1−lnx<0, or x>e≐2.7. (This is consistent with Figure 8.3.10(a).) Since the criteria of the test are satisfied, it follows that the series converges absolutely.
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