- Topics At a Glance
- Series: This is the Sum That Doesn't End
- Sigma Notation
- Alternating Series
- Convergence of Series
- Finally, Meaning...and Food
- Relationship Between Sequences
- Math-e-magics?
- When Limits of Summation Don't Matter
- Properties of Series
- Special Cases
- Arithmetic Series
- Geometric Series
- Finite Geometric Series
- Infinite Geometric Series
- Decimal Expansion
- Word Problems
- Visualization of Series
- When Limits of Summation Don't Matter
- Tests for Convergence
- The Divergence Test
- The Alternating Series Test
- The Ratio Test
- The Integral Test
- The Comparison Test
- Absolute Convergence vs. Conditional Convergence
- Summary of Tests
- Taylor and Maclaurin Series
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem
Introduction to :
Now that we know how to bake an infinite brownie, we want to know what it tastes like. We are going to need to find a cow big enough to give us a glass of milk to match.
In terms of series, this means adding up infinitely many numbers? We would expect the sum of an infinite list of numbers to be infinite. Some of the time, we'd be wrong.
Before we can understand how we can fit an infinite sum into a finite box, we have to define something called a partial sum. A partial sum is what we get if we add up some of the terms of a series. This is like breaking off a part of our infinite brownie and sharing it with friends.
More specifically, the nth partial sum of a series is the sum of the first n terms of the series.
Sample Problem
Find the 4th partial sum of the series
Answer.
To find the 4th partial sum we add the first 4 terms:
The 4th partial sum is 
.
The 1st partial sum of a series is its first term.
Sample Problem
The 1st partial sum of the series
is
If you think a series seems like the evil twin of a sequence, you are almost right. A series is the evil cousin of the sequence. They are related through a sequence of partial sums
S1, S2, S3, S4,... where the nth term is the nth partial sum of the series.
Be Careful: Remember that sequences and series are different things.
In reality, a series is the evil triplet of two different sequences. Any series
has two different sequences associated with it:
- The sequence whose terms are the same as the terms an of the series:
a1, a2, a3,...
- The sequence whose terms are the partial sums Sn of the series:
S1, S2, S3,...
These two sequences are different.
Practice:
Find S3 for the series 1 + 2 + 4 + 8 + 16 + .... |
Find S1 and S2 for the series
|
Find the sequence of partial sums for the series 1 + 2 + 4 + 8 + 16 + ... |
Find the nth partial sum of the series
for
(a) n = 1
(b) n = 3
(c) n = 5
Given the series
2 + 4 + 6 + 8 + 10 + ...
find
(a) S2
(b) S4
(c) S5
For the series, find the first 4 terms of the sequence of partial sums.
3 + 8 + 13 + 18 + ...
For the series, find the first 4 terms of the sequence of partial sums.
For the series, find the first 4 terms of the sequence of partial sums.
Fill in the blank with either the word 'sequence' or the word 'series.'
For any BLANK there is a BLANK of partial sums.
Fill in the blank with either the word 'sequence' or the word 'series.'
A partial sum is a BLANK.
Fill in the blank with either the word 'sequence' or the word 'series.'
A BLANK is a sum of numbers.
Fill in the blank with either the word 'sequence' or the word 'series.'
A BLANK can be either finite or infinite.
Fill in the blank with either the word 'sequence' or the word 'series.'
A list of numbers separated by commas is called a BLANK.
Fill in the blank with either the word 'sequence' or the word 'series.'
We can always think of a BLANK as being infinite.
