Chapter 8: Infinite Sequences and Series
Section 8.2: Series
Sum the series∑n=0∞1/3n and show that the sum is the limit of the sequence of partial sums.
The geometric series ∑n=0∞rn sums to S=11−r, and the finite sum Sk=∑n=0krn adds to 1−rk+11−r .
Since the given series is a geometric series with r=1/3, its sum is
The partial sum up through n=k is Sk=1−1/3k+11−1/3 , which, in the limit as k→∞, becomes 1−0 1−1/3=32.
Figure 8.2.1(a) shows the first few partial sums rapidly converging to S=3/2.
use plots in
Figure 8.2.1(a) Convergence of Sk to S
Maple "knows" how to sum a geometric series:
Obtain the sum of the series
Control-drag the given series.
Context Panel: Evaluate and Display Inline
∑n=0∞1/3n = 32
Obtain Sk and the first few partial sums
∑n=0k13n = −3⁢13k+12+32→sequence w.r.t. k1,43,139,4027,12181,364243,1093729,32802187,98416561,2952419683,8857359049
Obtain the limit of the sequence of partial sums
Calculus palette: Limit operator
Expression palette: Summation template
Sum the series to n=k.
limk→∞∑n=0k1/3n = 32
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