484
8.3. SERIES OF REAL NUMBERS
respectively. Explain why
A n + B n =
n
k=1
(a k + b k ).
(ii) Use the previous result and properties of limits to show that
∞
k=1
(a k + b k ) =
∞
k=1
a k +
∞
k=1
b k .
(Note that the starting point of the sum is irrelevant in this problem, so it
doesn’t matter where we begin the sum.)
(c) Use the prior result to find the sum of the series
∞
k=0
2 k +3 k
5 k .
4. In the Limit Comparison Test we compared the behavior of a series to one whose
behavior we know. In that test we use the limit of the ratio of corresponding terms of
the series to determine if the comparison is valid. In this exercise we see how we can
compare two series directly, term by term, without using a limit of sequence. First we
consider an example.
(a) Consider the series
1
k 2
and
1
k 2 + k
.
We know that the series
1
k 2 is a p-series with p = 2 > 1 and so
1
k 2
converges. In this part of the exercise we will see how to use information about
1
k 2 to determine information about
1
k 2 +k
. Let a k =
1
k 2 and b k =
1
k 2 +k
.
(i) Let S n be the nth partial sum of
1
k 2 and T n the nth partial sum of
1
k 2 +k
.
Which is larger, S 1 or T 1 ? Why?
(ii) Recall that
S 2 = S 1 + a 2
and
T 2 = T 1 + b 2 .
Which is larger, a 2 or b 2 ? Based on that answer, which is larger, S 2 or T 2 ?
(iii) Recall that
S 3 = S 2 + a 3
and
T 3 = T 2 + b 3 .
Which is larger, a 3 or b 3 ? Based on that answer, which is larger, S 3 or T 3 ?
(iv) Which is larger, a n or b n ? Explain. Based on that answer, which is larger,
S n or T n ?
(v) Based on your response to the previous part of this exercise, what relationship do you expect there to be between
1
k 2 and
1
k 2 +k
? Do you expect
8.3. SERIES OF REAL NUMBERS
respectively. Explain why
A n + B n =
n
k=1
(a k + b k ).
(ii) Use the previous result and properties of limits to show that
∞
k=1
(a k + b k ) =
∞
k=1
a k +
∞
k=1
b k .
(Note that the starting point of the sum is irrelevant in this problem, so it
doesn’t matter where we begin the sum.)
(c) Use the prior result to find the sum of the series
∞
k=0
2 k +3 k
5 k .
4. In the Limit Comparison Test we compared the behavior of a series to one whose
behavior we know. In that test we use the limit of the ratio of corresponding terms of
the series to determine if the comparison is valid. In this exercise we see how we can
compare two series directly, term by term, without using a limit of sequence. First we
consider an example.
(a) Consider the series
1
k 2
and
1
k 2 + k
.
We know that the series
1
k 2 is a p-series with p = 2 > 1 and so
1
k 2
converges. In this part of the exercise we will see how to use information about
1
k 2 to determine information about
1
k 2 +k
. Let a k =
1
k 2 and b k =
1
k 2 +k
.
(i) Let S n be the nth partial sum of
1
k 2 and T n the nth partial sum of
1
k 2 +k
.
Which is larger, S 1 or T 1 ? Why?
(ii) Recall that
S 2 = S 1 + a 2
and
T 2 = T 1 + b 2 .
Which is larger, a 2 or b 2 ? Based on that answer, which is larger, S 2 or T 2 ?
(iii) Recall that
S 3 = S 2 + a 3
and
T 3 = T 2 + b 3 .
Which is larger, a 3 or b 3 ? Based on that answer, which is larger, S 3 or T 3 ?
(iv) Which is larger, a n or b n ? Explain. Based on that answer, which is larger,
S n or T n ?
(v) Based on your response to the previous part of this exercise, what relationship do you expect there to be between
1
k 2 and
1
k 2 +k
? Do you expect