Cogito
AP Calculus BC · Chapter 3 · Lesson 1
Sequences and Partial Sums
A list, and the running total of that list.
11 problems · about 24 minutes · AP Calculus BC 10.1, 10.2
What this lesson teaches
The student distinguishes a sequence from a series and computes partial sums.
- 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.
Warm Up
Straightforward practice. Get the method working first.
4 problemsGeometric with a = 3, r = 1/2. What does it converge to?
Answer 6
Why 6.
1 + 3 + 5. Third partial sum?
Answer 9
Why Add them.
Geometric with a = 1, r = 1/3. What does it converge to? Give it as a decimal to two places.
Answer 1.5
Why 1 / (1 − 1/3).
Do the terms of the harmonic series go to zero? 1 for yes, 0 for no.
Answer 1
Why 1/n shrinks.
Build It Up
The same ideas with more to keep track of.
3 problemsDoes the harmonic series converge? 1 for yes, 0 for no.
Answer 0
Why Terms going to zero is not enough.
Sort each object by what it is.
Answer A sequence: 1, 2, 3, 4, …, The list of partial sums · A series: 1 + 2 + 3 + 4 + …
Why Plus signs make a series.
Geometric with a = 2, r = 1/2. Sum?
Answer 4
Why 2 / (1 − 1/2).
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsGeometric with r = 2. Does it converge? 1 for yes, 0 for no.
Answer 0
Why The ratio must be under 1 in size.
5 + 5 + 5. Third partial sum?
Answer 15
Why Three fives.
Three Terms: For 2 + 4 + 6, what is the third partial sum?
Answer 12
Why 12.
The Geometric Sum: 1 + 1/2 + 1/4 + … converges to what value?
Answer 2
Why 2.