Chapter 8: Infinite Sequences and Series
Section 8.2: Series
Obtain the sum of the series ∑n=1∞sin1n−sin1n+1 and show that the sum is the limit of the sequence of partial sums.
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 limk→∞Sk=sin1−limk→∞sin1k+1=sin1−0=sin1.
Obtain the sum of the series
Control-drag the series.
Context Panel: Evaluate and Display Inline
∑n=1∞sin1n−sin1n+1 = sin⁡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=1ksin1n−sin1n+1 = −sin⁡1k+1+sin⁡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
Context Panel: Conversions≻To List
Context Panel: Approximate≻5 (digits)
→sequence w.r.t. k
→at 5 digits
Obtain the limit of the partial sums
Calculus palette: Limit template≻Apply to Sk
limk→∞Sk = sin⁡1
Figure 8.2.8(a) shows the convergence of the first 15 members of the sequence of partial sums to S=sin1.
use plots in
Figure 8.2.8(a) Convergence of Sk to S=sin1
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