A sequence is an ordered list of numbers. A series is what you get by adding them.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Partial sums
The nth partial sum adds the first n terms. Those partial sums form a sequence of their own.
What convergence means
A series converges when its partial sums approach a limit — not when its terms do.
The trap
The terms of 1 + 1/2 + 1/3 + … go to zero, and the series still diverges.
A list and its running total
A sequence is an ordered list of terms; a series is the sum of those terms. The partial sums of a series form a new sequence, and the series converges exactly when that sequence of partial sums converges.
What convergence means
A sequence converges if its terms approach a single finite limit. A series converges if its partial sums do. These are different statements about different sequences, and the exam tests the distinction.
Terms going to zero is necessary, not sufficient
If a series converges, its terms must approach zero. The converse fails: the harmonic series has terms going to zero and diverges. This asymmetry is the source of a great many errors.
Sigma notation and indices
The index and its starting value both matter — shifting the start changes the sum. Writing out the first few terms of an unfamiliar series is the reliable way to see what it actually is.
Step 2: Try It Yourself
Tap and try it out.
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Step 3: Watch an Example
One step at a time.
Watch Omar Take Partial Sums
Omar adds the halving series 1 + 1/2 + 1/4 + 1/8.
- Step 1
The first partial sum is 1.
Step 4: Your Turn
Practice makes it stick.
Three Terms
Problem 1 of 2
For 2 + 4 + 6, what is the third partial sum?
The Geometric Sum
Problem 2 of 2
1 + 1/2 + 1/4 + … converges to what value?
Sums That Stop and Sums That Do Not
1 of 8
1 + 3 + 5. Third partial sum?
2 of 8
Geometric with a = 1, r = 1/3. What does it converge to? Give it as a decimal to two places.
3 of 8
Do the terms of the harmonic series go to zero? 1 for yes, 0 for no.
4 of 8
Does the harmonic series converge? 1 for yes, 0 for no.
5 of 8
Sort each object by what it is.
Tap something to move it.
- Empty
- Empty
6 of 8
Geometric with a = 2, r = 1/2. Sum?
7 of 8
Geometric with r = 2. Does it converge? 1 for yes, 0 for no.
8 of 8
5 + 5 + 5. Third partial sum?
Step 5: Quick Check
Show what you know.
Question 1 of 1
Geometric with a = 3, r = 1/2. What does it converge to?
What You Learned
- A sequence is a list; a series is the sum of that list.
- Convergence is about the partial sums, not the terms.
- Terms approaching zero is necessary but not sufficient.