498
8.4. ALTERNATING SERIES
approximating
π
4 with S 100 and explain why this is not a very good approximation.
(c) Determine the number of terms it would take in the series (8.17) to approximate
π
4 to 10 decimal places. (The fact that it takes such a large number of terms
to obtain even a modest degree of accuracy is why we say that conditionally
convergent series converge very slowly.)
2. We have shown that if
(−1) k+1 a k is a convergent alternating series, then the sum S
of the series lies between any two consecutive partial sums S n . This suggests that the
average
S n +S n+1
2
is a better approximation to S than is S n .
(a) Show that
S n +S n+1
2
= S n +
1
2 (−1) n+2 a n+1 .
(b) Use this revised approximation in (a) with n = 20 to approximate ln(2) given
that
ln(2) =
∞
k=1
(−1)
k+1 1
k
.
Compare this to the approximation using just S 20 . For your convenience,
S 20 =
155685007
232792560 .
3. In this exercise, we examine one of the conditions of the Alternating Series Test.
Consider the alternating series
1 − 1 +
1
2
−
1
4
+
1
3
−
1
9
+
1
4
−
1
16
+ · · · ,
where the terms are selected alternately from the sequences
1
n
and
−
1
n 2
.
(a) Explain why the nth term of the given series converges to 0 as n goes to infinity.
(b) Rewrite the given series by grouping terms in the following manner:
(1 − 1) +
1
2
−
1
4
+
1
3
−
1
9
+
1
4
−
1
16
+ · · · .
Use this regrouping to determine if the series converges or diverges.
(c) Explain why the condition that the sequence {a n } decreases to a limit of 0 is
included in the Alternating Series Test.
4. Conditionally convergent series exhibit interesting and unexpected behavior. In this exercise we examine the conditionally convergent alternating harmonic series
∞
k=1
(−1) k+1
k
and discover that addition is not commutative for conditionally convergent series. We
will also encounter Riemann’s Theorem concerning rearrangements of conditionally
8.4. ALTERNATING SERIES
approximating
π
4 with S 100 and explain why this is not a very good approximation.
(c) Determine the number of terms it would take in the series (8.17) to approximate
π
4 to 10 decimal places. (The fact that it takes such a large number of terms
to obtain even a modest degree of accuracy is why we say that conditionally
convergent series converge very slowly.)
2. We have shown that if
(−1) k+1 a k is a convergent alternating series, then the sum S
of the series lies between any two consecutive partial sums S n . This suggests that the
average
S n +S n+1
2
is a better approximation to S than is S n .
(a) Show that
S n +S n+1
2
= S n +
1
2 (−1) n+2 a n+1 .
(b) Use this revised approximation in (a) with n = 20 to approximate ln(2) given
that
ln(2) =
∞
k=1
(−1)
k+1 1
k
.
Compare this to the approximation using just S 20 . For your convenience,
S 20 =
155685007
232792560 .
3. In this exercise, we examine one of the conditions of the Alternating Series Test.
Consider the alternating series
1 − 1 +
1
2
−
1
4
+
1
3
−
1
9
+
1
4
−
1
16
+ · · · ,
where the terms are selected alternately from the sequences
1
n
and
−
1
n 2
.
(a) Explain why the nth term of the given series converges to 0 as n goes to infinity.
(b) Rewrite the given series by grouping terms in the following manner:
(1 − 1) +
1
2
−
1
4
+
1
3
−
1
9
+
1
4
−
1
16
+ · · · .
Use this regrouping to determine if the series converges or diverges.
(c) Explain why the condition that the sequence {a n } decreases to a limit of 0 is
included in the Alternating Series Test.
4. Conditionally convergent series exhibit interesting and unexpected behavior. In this exercise we examine the conditionally convergent alternating harmonic series
∞
k=1
(−1) k+1
k
and discover that addition is not commutative for conditionally convergent series. We
will also encounter Riemann’s Theorem concerning rearrangements of conditionally