Chapter 8: Infinite Sequences and Series
Section 8.2: Series
Use Maple to sum the series ∑n=1∞1n n+2 and show that the sum is the limit of the sequence of partial sums.
A partial fraction decomposition gives 1n n+1=121n−1n+2, so that the given series is a telescoping series. (See Table 8.2.2.)
Twice the kth partial sum is then
where the red terms cancel in pairs, but the two bold terms will be left in any finite sum.
Consequently, the sum of the series is given by
Obtain the sum of the series
Control-drag the series.
Context Panel: Evaluate and Display Inline
∑n=1∞1n n+2 = 34
Obtain an expression for the kth partial sum
Control-drag the series and change ∞ to k.
Context Panel: Assign to a Name≻S[k]
∑n=1k1n n+2 = −12⁢k+1−12⁢k+2+34→assign to a nameSk
Display the first few partial sums
Type Sk and press the Enter key.
Context Panel: Sequence≻k
In the resulting dialog box, set k=1 to k=15
→sequence w.r.t. k
Obtain the limit of the partial sums
Calculus palette: Limit template≻Apply to Sk
limk→∞Sk = 34
Figure 8.2.11(a) shows the convergence of the first 15 members of the sequence of partial sums to S=3/4.
use plots in
Figure 8.2.11(a) Convergence of Sk to S=3/4
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