Chapter 8: Infinite Sequences and Series
Section 8.2: Series
Sum the series ∑n=1∞1n n+1 and show that the sum is the limit of the sequence of partial sums.
A partial fraction decomposition gives 1n n+1=1n−1n+1, so that the given series is a telescoping series. (See Table 8.2.2.)
The kth partial sum is then
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+1 = 1
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+1 = −1k+1+1→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 = 1
Figure 8.2.4(a) shows the convergence of the first 15 members of the sequence of partial sums to S=1.
use plots in
Figure 8.2.4(a) Convergence of Sk to S=1
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