1. Check the Ratio ($r$): A series only has a limit if $|r| < 1$.
2. Identify $a_1$: This is your starting point.
3. The Sum Formula: $S_{\infty} = a_1 / (1 - r)$.
4. Limit at Infinity: As the bottom of a fraction gets infinitely big, the whole fraction gets infinitely small (zero).
Mark each series as either **Convergent** (has a limit) or **Divergent** (explodes to infinity).
Series A: $10 + 5 + 2.5 + 1.25...$
Series B: $1 + 3 + 9 + 27...$
Series C: $100 - 50 + 25 - 12.5...$
Find the exact limit of these infinite geometric series.
The Bread: $1/2 + 1/4 + 1/8 + 1/16...$
The Step: You take a step of 6 feet. Your next step is 2 feet. The next is 2/3 feet.
If you do this forever, how far will you travel?
If you have an infinite series where $r = 1$, like $5 + 5 + 5 + 5...$ is it convergent or divergent? Why can't we find a "Limit" for this sum?
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Evaluate these limits as $x$ approaches infinity ($\infty$).
$\lim_{x \to \infty} (10 / x)$
$\lim_{x \to \infty} (\frac{2x + 5}{x})$
A ball is dropped from a height of 20 meters. It bounces back up to 75% of its previous height ($r=0.75$).
1. Calculate the infinite sum of all the **Upward** bounces ($a_1 = 15$).
2. Calculate the infinite sum of all the **Downward** paths ($a_1 = 20$).
3. Add them together to find the total distance the ball travels before it stops.
Objective: Explain the limit to a younger student using food.
The Activity:
1. Take a cookie. Eat half.
2. Give them half of what's left.
3. Take half of what's left.
4. Ask: "If we do this all day, will we ever eat more than just this one cookie?"
The Lesson: "Infinite pieces can still fit inside one whole. That's how God fits His infinite love inside our one heart."
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