37.27
Evaluate
is the harmonic series minus the first 99 terms. However, convergence or divergence is not
affected by deletion or addition of any finite number of terms. Since the harmonic series is divergent (by
Problem 37.2), so is the given series.
INFINITE SERIES
315
37.28
Evaluate
Since the harmonic series is divergent, so is the given series.
37.29
37.30
37.31
37.32
37.33
37.34
Evaluate
In [nl(n + 1)] = In n - In (n + 1),
and
5,, = (In 1 - In 2) + (In 2 - In 3) + - • • + [In n - In (n + 1)] =
-In (« + !)-» -oo. Thus, the given series diverges.
Evaluate
This is a geometric series with ratio r=\le<\ and first term a = l. Hence, it converges to
Evaluate
This series converges because it is the sum of two convergent series,
and
(both are
geometric series with ratio
and
Hence, the sum of the
given series is 2 + f = ".
(Zeno's paradox) Achilles (A) and a tortoise (7") have a race. T gets a 1000-ft head start, but A runs at 10 ft/s
while the tortoise only does 0.01 ft/s. When A reaches T's starting point, T has moved a short distance ahead.
When A reaches that point, T again has moved a short distance ahead, etc. Zeno claimed that A would never
catch T. Show that this is not so.
When A reaches T's starting point, 100 s have passed and T has moved 0.01 x 100 = 1 ft. A covers that
additional 1ft in O.ls, but T has moved 0.01 x 0.1 =0.001 ft further. A needs 0.0001 s to cover that distance,
but T meanwhile has moved 0.01 x 0.0001 = 0.000001 ft; and so on. The limit of the distance between A and
T approaches 0. The time involved is 100 + 0.1 + 0.0001 + 0.0000001 + • • • , which is a geometric series with
first term a = 100 and ratio r=-^. Its sum is 100/(1 - Tm). Thus, Achilles catches up with (and then
passes) the tortoise in a little over 100 s, just as we knew he would. The seeming paradox arises from the artificial
division of the event into infinitely many shorter and shorter steps.
A rubber ball falls from an initial height of 10m; whenever it hits the ground, it bounces up two-thirds of the
previous height. What is the total distance covered by the ball before it comes to rest?
The distance is 10 + 2[10(§) + 10(|)
? + 10(|)
3 + • • •]. In brackets is a geometric series with ratio f and first
term f; its sum is f /(I - §) = 20, for a distance of 10 + 2(20) = 50 m.
Investigate
3/(5"=1/5n-1.So this series of positive terms is term by term less than the convergent
geometric series
Hence, by the comparison test, the given series is convergent. However, we
cannot directly compute the sum of the series. We can only say that the sum is less than
Evaluate
is the harmonic series minus the first 99 terms. However, convergence or divergence is not
affected by deletion or addition of any finite number of terms. Since the harmonic series is divergent (by
Problem 37.2), so is the given series.
INFINITE SERIES
315
37.28
Evaluate
Since the harmonic series is divergent, so is the given series.
37.29
37.30
37.31
37.32
37.33
37.34
Evaluate
In [nl(n + 1)] = In n - In (n + 1),
and
5,, = (In 1 - In 2) + (In 2 - In 3) + - • • + [In n - In (n + 1)] =
-In (« + !)-» -oo. Thus, the given series diverges.
Evaluate
This is a geometric series with ratio r=\le<\ and first term a = l. Hence, it converges to
Evaluate
This series converges because it is the sum of two convergent series,
and
(both are
geometric series with ratio
and
Hence, the sum of the
given series is 2 + f = ".
(Zeno's paradox) Achilles (A) and a tortoise (7") have a race. T gets a 1000-ft head start, but A runs at 10 ft/s
while the tortoise only does 0.01 ft/s. When A reaches T's starting point, T has moved a short distance ahead.
When A reaches that point, T again has moved a short distance ahead, etc. Zeno claimed that A would never
catch T. Show that this is not so.
When A reaches T's starting point, 100 s have passed and T has moved 0.01 x 100 = 1 ft. A covers that
additional 1ft in O.ls, but T has moved 0.01 x 0.1 =0.001 ft further. A needs 0.0001 s to cover that distance,
but T meanwhile has moved 0.01 x 0.0001 = 0.000001 ft; and so on. The limit of the distance between A and
T approaches 0. The time involved is 100 + 0.1 + 0.0001 + 0.0000001 + • • • , which is a geometric series with
first term a = 100 and ratio r=-^. Its sum is 100/(1 - Tm). Thus, Achilles catches up with (and then
passes) the tortoise in a little over 100 s, just as we knew he would. The seeming paradox arises from the artificial
division of the event into infinitely many shorter and shorter steps.
A rubber ball falls from an initial height of 10m; whenever it hits the ground, it bounces up two-thirds of the
previous height. What is the total distance covered by the ball before it comes to rest?
The distance is 10 + 2[10(§) + 10(|)
? + 10(|)
3 + • • •]. In brackets is a geometric series with ratio f and first
term f; its sum is f /(I - §) = 20, for a distance of 10 + 2(20) = 50 m.
Investigate
3/(5"=1/5n-1.So this series of positive terms is term by term less than the convergent
geometric series
Hence, by the comparison test, the given series is convergent. However, we
cannot directly compute the sum of the series. We can only say that the sum is less than