Chapter 8: Infinite Sequences and Series
Section 8.3: Convergence Tests
Determine if the series ∑n=1∞cosnn2 diverges, converges absolutely, or converges conditionally.
If it converges conditionally, determine if it also converge absolutely.
Because cosn is sometimes positive and sometimes negative, but not in strict alternating sequence, this is a series with both positive and negative terms, but it is not an alternating series. Hence, the Leibniz test cannot be used.
Instead, establish the absolute convergence (and hence, the conditional convergence) of the series by applying the Comparison test, comparing the series with the convergent p-series Σ 1/n2. Indeed, cosn/n2≤1/n2, so the given series converges absolutely, and hence, conditionally.
Maple sums this series in terms of the special function polylog, thus establishing its conditional convergence. Absolute convergence is established by the Comparison test, as shown in the Mathematical Solution, above.
Expression palette: Summation template
Context Panel: Evaluate and Display Inline
Context Panel: Approximate≻5 (digits)
∑n=1∞cosnn2 = 12⁢polylog⁡2,ⅇI+12⁢polylog⁡2,ⅇ−I→at 5 digits0.32414+0.⁢I
The subtraction of "0.I" from the positive real number 0.32414 is an artifact of the evaluation of the polylog function. The series sums exactly to a real number.
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