Chapter 8: Infinite Sequences and Series
Section 8.3: Convergence Tests
Determine if the series ∑n=1∞arctannn2+4 diverges, converges absolutely, or converges conditionally.
If it converges conditionally, determine if it also converge absolutely.
Since arctann→π/2 as n→∞, it would seem that comparing this series to the convergent p-series ∑n=1∞1n2 might be a useful ploy, in which case consider
By part (1) of the Limit-Comparison test (Table 8.3.1), both series either converge or diverge. But the p-series converges, so the given series converges. Since the given series has only positive terms, the convergence is absolute.
As per the Mathematical Solution above, the absolute convergence of the given series can be established by the Limit-Comparison test, by a comparison to the convergent p-series ∑n=1∞1n2. Absolute convergence can also be established by the Integral test, but the integration is not elementary.
Figure 8.3.1(a) contains a graph of the function fx=arctanxx2+4 (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
Context Panel; Approximate≻5 (digits)
Figure 8.3.1(a) Graph of fx (red) and f′x (green)
→at 5 digits
The "0.I" appended to 0.53317 is an artifact of the numeric approximation of the exact answer Maple finds for this integral. In essence, the value of the integral is a real, positive number, and by the terms of the Integral test, the given series converges absolutely.
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