(a) The 1st partial sum of a series is its first term. In this case, the 1st partial sum is
2(1) = 2.
(b) The 3rd partial sum is the sum of the first 3 terms:
2(1) + 2(2) + 2(3) = 12.
(c) The 5th partial sum is the sum of the first 5 terms:
2(1) + 2(2) + 2(3) + 2(4) + 2(5) = 30.
We usually denote the nth partial sum of a series by Sn.
Example 2
Given the series
2 + 4 + 6 + 8 + 10 + ...
find
(a) S2
(b) S4
(c) S5
Answer
(a) S2 is the 2nd partial sum, which is the sum of the first 2 terms.
S2 = 2 + 4 = 6.
(b) S4 is the 4th partial sum, which is the sum of the first 4 terms.
S4 = 2 + 4 + 6 + 8 = 20
(c) S5 is the 5th partial sum, which is the sum of the first 5 terms.
S5 = 2 + 4 + 6 + 8 + 10 = 30.
Example 3
For the series, find the first 4 terms of the sequence of partial sums.
3 + 8 + 13 + 18 + ...
Answer
The first four partial sums are
S1 = 3 = 3
S2 = 3 + 8 = 11
S3 = 3 + 8 + 13 = 24
S4 = 3 + 8 + 13 + 18 = 42
The first 4 terms of the sequence of partial sums are
3, 11, 24, 42.
Example 4
For the series, find the first 4 terms of the sequence of partial sums.
Answer
The first 4 partial sums are
This means the first 4 terms of the sequence of partial sums are
Example 5
For the series, find the first 4 terms of the sequence of partial sums.
Answer
Ignore the fact that the indexing starts at 0. If we expand the summation notation, we get
0 + 1 + 4 + 9 + 16 + ...
The first term of the series is 0. We need to account for that when we find the partial sums.
S1 = 0 = 0
S2 = 0 + 1 = 1
S3 = 0 + 1 + 4 = 5
S4 = 0 + 1 + 4 + 9 = 14
The first 4 terms of the sequence of partial sums are
0, 1, 5, 14
Example 6
Fill in the blank with either the word 'sequence' or the word 'series.'
For any BLANK there is a BLANK of partial sums.
Answer
For any SERIES there is a SEQUENCE of partial sums.
Example 7
Fill in the blank with either the word 'sequence' or the word 'series.'
A partial sum is a BLANK.
Answer
A partial sum is a SERIES.
Example 8
Fill in the blank with either the word 'sequence' or the word 'series.'
A BLANK is a sum of numbers.
Answer
A SERIES is a sum of numbers.
Example 9
Fill in the blank with either the word 'sequence' or the word 'series.'
A BLANK can be either finite or infinite.
Answer
Either one works. A SEQUENCE can be either finite or infinite. A SERIES can be either finite or infinite. However, remember that we can always
think of a series as being infinite.
Example 10
Fill in the blank with either the word 'sequence' or the word 'series.'
A list of numbers separated by commas is called a BLANK.
Answer
A list of numbers separated by commas is called a SEQUENCE.
Example 11
Fill in the blank with either the word 'sequence' or the word 'series.'
We can always think of a BLANK as being infinite.
Answer
We can always think of a SERIES as being infinite. We can't always think of a sequence as being infinite.
