Chapter 8: Infinite Sequences and Series
Section 8.3: Convergence Tests
Determine if the series ∑n=0∞3n3 n+2! diverges, converges absolutely, or converges conditionally.
If it converges conditionally, determine if it also converge absolutely.
This series of positive terms is absolutely convergent, as shown by the Ratio test, based on the following calculation.
=limn→∞3n+13 n+1+2!3n3 n+2!
=limn→∞33 n+33 n+43 n+5
Since L is less than 1, the series converges absolutely.
Define an as a function of n
Context Panel: Assign Function
an=3n/3 n+2!→assign as functiona
Apply the Ratio test by calculating L
Calculus palette: Limit template
Context Panel: Evaluate and Display Inline
limn→∞an+1an = 0
Obtain the sum of the series
Expression palette: Summation template
Press the Enter key.
Context Panel: Approximate≻5 (digits)
→at 5 digits
Showing that this series of positive terms has an explicit sum would actually establish its absolute convergence, provided that Maple's internal manipulations were exposed. Short of that, either the details of the summation would need to be provided, or the Ratio test implemented.
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