Sum of geometric series (AA SL 1.8, AI HL 1.11)
Keywords
[table][br][tr][br][td]English[/td][br][td]Japanese[/td][br][td]Korean[/td][br][td]Chinese Simplified[/td][br][/tr][br][tr][br][td]Infinite Geometric Series[/td][br][td]無限等比級数[/td][br][td]무한 등비급수[/td][br][td]无限等比级数[/td][br][/tr][br][tr][br][td]Sum Formula[/td][br][td]和の公式[/td][br][td]합 공식[/td][br][td]求和公式[/td][br][/tr][br][tr][br][td]Convergence[/td][br][td]収束[/td][br][td]수렴[/td][br][td]收敛[/td][br][/tr][br][tr][br][td]Common Ratio[/td][br][td]公比[/td][br][td]공비[/td][br][td]公比[/td][br][/tr][br][tr][br][td]Divergence[/td][br][td]発散[/td][br][td]발산[/td][br][td]发散[/td][br][/tr][br][tr][br][td]Asymptotic Behavior[/td][br][td]漸近的な振る舞い[/td][br][td]점근적 행동[/td][br][td]渐近行为[/td][br][/tr][br][tr][br][td]Sum to Infinity[/td][br][td]無限大の和[/td][br][td]무한대의 합[/td][br][td]无穷和[/td][br][/tr][br][tr][br][td]Geometric Progression[/td][br][td]等比数列[/td][br][td]등비수열[/td][br][td]等比数列[/td][br][/tr][br][tr][br][td]Limit[/td][br][td]極限[/td][br][td]한계[/td][br][td]极限[/td][br][/tr][br][/table][br]
[table][br][tr][br][td][b]Factual Questions[/b][/td][br][td][b]Conceptual Questions[/b][/td][br][td][b]Debatable Questions[/b][/td][br][/tr][br][tr][br][td]1. What is the formula for the sum of an infinite geometric series?[/td][br][td]1. Why does the sum of an infinite geometric series only converge for |r| < 1?[/td][br][td]1. Is the concept of infinite geometric series more abstract than that of finite geometric series?[/td][br][/tr][br][tr][br][td]2. Calculate the sum of the infinite series 0.5 + 0.25 + 0.125 + ...[/td][br][td]2. Explain how the formula for the sum of an infinite geometric series is derived.[/td][br][td]2. Can the concept of infinite geometric series be effectively applied in real-world scenarios?[/td][br][/tr][br][tr][br][td]3. Under what condition does an infinite geometric series converge?[/td][br][td]3. Discuss the concept of convergence and divergence in the context of geometric series.[/td][br][td]3. Debate the importance of understanding infinite series in the broader context of mathematics.[/td][br][/tr][br][tr][br][td]4. Find the sum of the infinite geometric series 3 + 1.5 + 0.75 + ...[/td][br][td]4. How does changing the first term of an infinite geometric series affect its sum?[/td][br][td]4. Discuss the statement: "The sum of an infinite series challenges our conventional understanding of infinity."[/td][br][/tr][br][tr][br][td]5. What is the common ratio in the series 2 + 6 + 18 + 54 + ...?[/td][br][td]5. Compare the behavior of convergent and divergent geometric series.[/td][br][td][/td][br][/tr][br][/table][br]
Mini-Investigation: The Geometric Series JourneyWelcome to the Geometric Series Journey, a numerical adventure to understand the power of multiplication and addition! Let's dive into the realm of geometric progression with a playful exploration.[br][br]Chart your discoveries and share them with other math explorers.
1. Starting Strong: Our series starts with a first term, [math]U₁[/math], of 50 and a common ratio, [math]r[/math], of [math]0.9[/math]. What happens to the terms of the series as we progress? Predict the 10th term without peeking and then check!
2. Ratio Riddles: Change the common ratio to [math]1.1[/math]. How does this affect the progression of our series? Do the terms increase or decrease? Test your hypothesis and observe the new graph.
3. Summing It Up: Look at the summation table. Notice how the sum, Sₙ, increases as n gets larger. What do you think the sum would be after [math]20[/math] terms? After [math]50[/math] terms? Take a guess and then use the applet to find out.
4. Asymptotic Adventures: The graph seems to approach a horizontal dashed line as n increases. Can you guess the value it's approaching? Hint: There’s a formula for the sum of an infinite geometric series!
5. Break the Bank: Suppose you start with 50$ and keep adding 90% of the previous amount you added. How much money will you have after 10 additions? Does this match with the applet's Sₙ for n=10?
6. Limit Quest: If the common ratio is between -1 and 1, the series has a limit. What happens to the series when the common ratio is outside this range? Experiment and document the results.
7. Common Ratio Conundrum: What if the first term stays the same, but you can choose any common ratio? Find a ratio that gives you a sum as close to 500 as possible without going over, in the first 10 terms.
8. Negative Notions: Change the common ratio to a negative value, like -0.9. How does the sign of the ratio affect the series? Observe the graph and summarize your findings.
9. Geometric Growth: How would you describe the growth of a geometric series to a friend who’s never seen one before? Think of a real-world example that illustrates this concept.
10. Infinity Inquiry: The applet allows us to see the sum of finite terms. But what does it mean to sum an infinite geometric series? Can you have an infinite sum that’s a finite number?
Part 2 - Checking for understanding[br][br]Watch the below video before attempting the quiz questions
Link to OSC - Oxford Study Courses youtube channel https://www.youtube.com/@OSC1990
Does the sum to infinity exist for the series [math]3,6,12,\dots[/math]?
Does the sum to infinity exist for the series [math]100,50,25,\dots?[/math]
Does the sum to infinity exist for the series [math]5,−2.5,1.25,\dots[/math]?
What is the sum to infinity of the series 60,30,15,…
Calculate the sum to infinity of the series 100,50,25,…?
Find the sum to infinity of the series 40,−20,10,....?
Question: What is the sum to infinity of the series 5,10,20,…?
Lesson Plan- Understanding the Sum of Infinite Geometric Series
Sum of geometric series- Intuition pump (thought experiments and analogies)
