Calculus 2nd Edition Rogawski

SECTION 11.3 Convergence of Series with Positive Terms 565
(b) Show, by integrating over [0, 1], that
π
4 = 1 − 1 3 + 1 5 − 1 ∫
(−1)N−1
1
+···+
7 2N − 1 + x 2N dx
(−1)N
0 1 + x 2
(c) Use the Comparison Theorem for integrals to prove that
than L/4 on the table. It will not fall through because it will not fit
through any of the removed sections.
L/16 L/4 L/16
∫ 1 x 2N dx
0 ≤
0 1 + x 2 ≤ 1
2N + 1
Hint: Observe that the integrand is ≤ x 2N .
(d) Prove that
π
4 = 1 − 1 3 + 1 5 − 1 7 + 1 9 − ···
Hint: Use (b) and (c) to show that the partial sums S N of satisfy
∣ SN − π ∣
4 ≤
1
2N+1 , and thereby conclude that lim S N = π
N→∞
4 .
54. Cantor’s Disappearing Table (following Larry Knop of Hamilton
College) Take a table of length L (Figure 7). At stage 1, remove the
section of length L/4 centered at the midpoint. Two sections remain,
each with length less than L/2. At stage 2, remove sections of length
L/4 2 from each of these two sections (this stage removes L/8 of the
table). Now four sections remain, each of length less than L/4. At stage
3, remove the four central sections of length L/4 3 , etc.
(a) Show that at the Nth stage, each remaining section has length less
than L/2 N and that the total amount of table removed is
( 1
L
4 + 1 8 + 1 16 +···+ 1 )
2 N+1
(b) Show that in the limit as N →∞, precisely one-half of the table
remains.
This result is curious, because there are no nonzero intervals of table left
(at each stage, the remaining sections have a length less than L/2 N ). So
the table has “disappeared.” However, we can place any object longer
FIGURE 7
55. The Koch snowflake (described in 1904 by Swedish mathematician
Helge von Koch) is an infinitely jagged “fractal” curve obtained
as a limit of polygonal curves (it is continuous but has no tangent line
at any point). Begin with an equilateral triangle (stage 0) and produce
stage 1 by replacing each edge with four edges of one-third the length,
arranged as in Figure 8. Continue the process: At the nth stage, replace
each edge with four edges of one-third the length.
(a) Show that the perimeter P n of the polygon at the nth stage satisfies
P n = 4 3 P n−1. Prove that lim
n→∞ P n = ∞. The snowflake has infinite
length.
(b) Let A 0 be the area of the original equilateral triangle. Show that
(3)4 n−1 new triangles are added at the nth stage, each with area A 0 /9 n
(for n ≥ 1). Show that the total area of the Koch snowflake is 8 5 A 0.
Stage 1 Stage 2 Stage 3
FIGURE 8
3
2
y
1
a 1 a 2 a 3 a N x
1 2 3 N
FIGURE 1 The partial sum S N is the sum of
the areas of the N shaded rectangles.
11.3 Convergence of Series with Positive Terms
The next three sections develop techniques for determining whether an infinite series
converges or diverges. This is easier than finding the sum of an infinite series, which is
possible only in special cases.
In this section, we consider positive series ∑ a n , where a n > 0 for all n. We can
visualize the terms of a positive series as rectangles of width 1 and height a n (Figure 1).
The partial sum
S N = a 1 + a 2 +···+a N
is equal to the area of the first N rectangles.
The key feature of positive series is that their partial sums form an increasing sequence:
S N