Convergence Lab Worksheet Convergence Lab
Topic: Series Efficiency & Euler's Number
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Phase 01
Factorial Foundation
Before we dive into the infinite, let's brush up on our notation. Calculate the following values:
Definition of 0!
Calculate 4!
Calculate 5!
Phase 02
The Factorial Series
Watch the video segment (7:08-9:30). The narrator expresses \(e\) as an infinite sum of reciprocal factorials. Note the progression below:
Term Index (\(n\)) Fractional Term (\(1/n!\)) Decimal Value (approx) 0 1/0! 1.0000 1 1/1! 1.0000 2 1/2! 0.5000 3 1/3! 0.1666... 4 1/4! ... ... ... 8 1/8!
Final Sum (\(S_8\)) shown in video:
Phase 03
Series Efficiency
Using your graphing calculator, calculate the partial sums \(S_n = \sum_{k=0}^{n} \frac{1}{k!}\) and the error relative to the actual value of \(e \approx 2.718281828\).
| \(n\) | Sum \(S_n\) | Error \(|e - S_n|\) |
| --- | --- | --- |
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
Error Magnitude
Number of Terms (n)
Phase 04
Comparative Convergence
Analysis: Limit vs. Series
In the video (4:05), the limit method required \(n = 1,000,000\) to reach 5 decimal places of accuracy (\(2.71828\)). Looking at your table from Phase 03, how many terms (\(n\)) did the factorial series require to reach that same level of accuracy? Why do factorials "win"?
The Grand Connection
We've seen that when we add up reciprocal factorials, we get \(e\). This is actually a specific case of a Taylor Series . The power series expansion for \(f(x) = e^x\) centered at \(c = 0\) is:
\[e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots\]
Prove that the video's formula is correct by substituting \(x = 1\) into the general series above:
Factorial Fast Track • AP Calculus BC • Convergence Lab Unit
Euler Engine Slides AP Calculus BC
Euler's Engine
Exploring the Power and Efficiency of the Infinite Factorial Series for \(e\).
CONVERGENCE LAB 01
The Power of "!"
Phase 01: Warm-up
Factorials grow unbelievably fast . This speed is the secret weapon of the series for \(e\).
0! = 1
4! = 24
5! = 120
Think-Pair-Share
"If factorials are in the denominator, what does their rapid growth imply about the size of the terms in the series as n increases?"
The Series Definition
7:08 - 9:30
Embedded media
"Which method seems to get to the value of \(e\) faster: the compound interest limit or the factorial sum? Why?"
Series Efficiency Lab
Phase 03: Activity
Your Mission
Calculate the first 5 partial sums of the series.
Find the Absolute Error for each sum compared to actual \(e\).
Graph the Error Magnitude vs. \(n\).
Calculator Tip
sum(seq(1/K!, K, 0, N))
Get Ready to Crunch Numbers
1/0! 1/1! 1/2! 1/3! 1/4! 1/5! 1/6!
The Big Picture
Phase 04: Closure
The series from the video is just one slice of the Taylor Series :
\[e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}\]
When \(x = 1\)
1 + 1 + 1/2 + 1/6 + ... = e
When \(x = 2\)
1 + 2 + 4/2 + 8/6 + ... = e²
Calculator Toolkit Reference Calculator Toolkit
Series & Sequences Reference
REF-CODE: CALC-BC-EULER
Efficiency isn't just about the series—it's about how you use your tools. Use these commands to streamline the calculations for the Convergence Lab activity.
Method 1: The Sequence Sum
TI-84 / TI-84 Plus CE Commands
STEP 1
Press 2nd then STAT (LIST). Arrow over to MATH and select 5: sum(.
STEP 2
Press 2nd then STAT (LIST) again. Arrow over to OPS and select 5: seq(.
Final Syntax
sum(seq(1/K!, K, 0, N))
Expression
Variable
Start
End (n)
Calculating Relative Error
The Formula
|e - Sn|
Subtract your partial sum from the calculator's built-in constant \(e\).
Pro Tip: Storage
Store the summation as a function to quickly iterate through \(n\) values.
Y1 = sum(seq(1/K!, K, 0, X))
Use the TABLE feature (2nd GRAPH) to see error for all X values at once!
Visualizing Convergence
Window Settings for Graphing Error
Xmin: 0
Xmax: 10
Xscl: 1
Ymin: 0
Ymax: 0.5
Yscl: 0.1
Adjust Ymax downward as you increase N to see the "vanishing" error.
Convergence Lab Toolkit AP Calculus BC Generated 2026
Convergence Lab Answer Key Answer Key & Teacher Guide
Convergence Lab: Factorial Fast Track
Confidential
01
Factorial Foundation
0!
1
4!
24
5!
120
02
Video Notes (7:08-9:30)
n Term Decimal Value 4 1/24 0.041666... ... ... ... 8 1/40320 0.0000248...
Video Sum (\(S_8\)): 2.7182787... (Highly accurate after just 9 terms!)
03
Series Efficiency Table (\(e \approx 2.7182818\))
| n | Sum \(S_n\) | Error \(|e - S_n|\) |
| --- | --- | --- |
| 1 | 2.0000000 | 0.7182818 |
| 2 | 2.5000000 | 0.2182818 |
| 3 | 2.6666667 | 0.0516151 |
| 4 | 2.7083333 | 0.0099485 |
| 5 | 2.7166667 | 0.0016151 |
Graphing Note:
Students should produce a rapidly decaying exponential-like curve . The error drops by approximately a factor of \(1/n\) for each step.
04
Analysis & Proof
Limit vs. Series Analysis
"The limit method required \(1,000,000\) iterations to reach \(2.71828\). The factorial series reached that level of accuracy (approx. 0.0001 error) at roughly \(n=7\) or \(n=8\). Factorials 'win' because they grow super-exponentially (\(n!\)), meaning the terms in the series (\(1/n!\)) shrink much faster than the terms in the limit expression."
Taylor Series Connection (x = 1)
\[e^x = \frac{x^0}{0!} + \frac{x^1}{1!} + \frac{x^2}{2!} + \dots\] \[e^1 = \frac{1^0}{0!} + \frac{1^1}{1!} + \frac{1^2}{2!} + \dots\] \[e = 1 + 1 + \frac{1}{2} + \frac{1}{6} + \dots\]
This perfectly matches the summation \(\sum \frac{1}{n!}\) shown in the video.
Teacher Reference Guide • Factorial Fast Track • AP Calculus BC
