Chapter 8: Infinite Sequences and Series
Section 8.3: Convergence Tests
Example 8.3.2
Determine if the series ∑n=2∞1n lnn diverges, converges absolutely, or converges conditionally.
If it converges conditionally, determine if it also converge absolutely.
Solution
Divergence of this series can be established by the Integral test.
Figure 8.3.2(a) contains a graph of the function fx=1x lnx (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≥2. (The derivative appears to be negative for x>2.)
Consequently, the Integral test may be tried, provided the integration starts from, say, x=2. This is a nontrivial integration, one Maple evaluates in terms of the special function dilogx.
To this end:
Calculus palette: Definite integral template Context Panel: Evaluate and Display Inline
∫2∞1x lnx ⅆx = ∞
module() local F,p; F:=1/x/ln(x); p:=plot([F,diff(F,x)],x=1..5,color=[red,green],view=[0..5,-10..10],tickmarks=[5,default]); print(p); end module:
Figure 8.3.2(a) Graph of fx (red) and f′x (green)
Since the integral is unbounded, the series diverges.
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