Cogito
AP Calculus BC · Chapter 3 · Lesson 2
Geometric and Telescoping Series
Two series whose sums can be found exactly.
12 problems · about 21 minutes · LIM-7.A
What this lesson teaches
The student determines convergence and sums of geometric and telescoping series.
- A geometric series converges exactly when |r| < 1, to a ÷ (1 − r).
- a is the first term of the series as written.
- A telescoping series is summed through its partial sum.
Warm Up
Straightforward practice. Get the method working first.
5 problemsFirst term 10, ratio 0.5. What is the infinite sum?
Answer 20
Why 20.
What is the method for a telescoping series?
Answer Find a closed form for the partial sum, then take its limit.
Why The partial sum and its limit.
First term 4, ratio 0.5. Infinite sum?
Answer 8
Why 4 ÷ 0.5.
First term 1, ratio 0.8. Infinite sum?
Answer 5
Why 1 ÷ 0.2.
First term 3, ratio 0.25. Infinite sum?
Answer 4
Why 3 ÷ 0.75.
Build It Up
The same ideas with more to keep track of.
3 problemsRatio 1. Does the series converge? 1 yes, 0 no.
Answer 0
Why The terms never shrink.
Ratio −0.5. Does it converge? 1 yes, 0 no.
Answer 1
Why The size is below 1.
First term 20, ratio 0.5. Infinite sum?
Answer 40
Why 20 ÷ 0.5.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsWhich ratios give a convergent geometric series?
Answer r = 0.3; r = negative 0.5
Why The size must be strictly below 1.
First term 6, ratio 0.5. Infinite sum?
Answer 12
Why 6 ÷ 0.5.
The Sum: First term 8, ratio 0.5. What is the infinite sum?
Answer 16
Why 16.
The Divergent One: Ratio 2. Does the series converge? 1 yes, 0 no.
Answer 0
Why No.