Sequences and Series

Infinite Series: The Magic of Adding Forever 🌟

The Story of the Stacking Coins

Imagine you have a magical piggy bank. Every day, you add a coin. Day 1, you add 1 coin. Day 2, another coin. Day 3, another. You keep going… forever!

But here’s the twist: what if the coins you add get smaller each time? What if by day 100, you’re adding coins tinier than a grain of sand? Does your piggy bank overflow, or does it stay just right?

This is the heart of Infinite Series. Let’s discover together!

📖 What is a Sequence?

A sequence is just a list of numbers in a specific order.

Think of it like waiting in line for ice cream:

• Person 1, Person 2, Person 3… and so on.

Each person has a position (1st, 2nd, 3rd…) and each position has a value (the person’s name or number).

Simple Example:

Sequence: 2, 4, 6, 8, 10, ...

• 1st term: 2
• 2nd term: 4
• 3rd term: 6
• Pattern: Add 2 each time!

The Formula Way:

We write the nth term as: aₙ = 2n

That’s a sequence! A recipe that tells you what number goes in each position.

🎯 Sequence Limits: Where Does the Line End?

Now here’s where it gets magical!

Some sequences keep growing forever (like 1, 2, 3, 4, 5…). But some sequences approach a destination without ever quite reaching it.

The Shrinking Steps Story

Imagine you’re walking toward a wall. Each step, you walk half the remaining distance.

• Step 1: You’re at 1/2 the way
• Step 2: You’re at 3/4 the way
• Step 3: You’re at 7/8 the way
• Step 4: You’re at 15/16 the way

You get closer and closer to 1 (the wall), but never touch it!

Example Sequence:

aₙ = 1/n

1, 1/2, 1/3, 1/4, 1/5, ...

As n gets HUGE, what happens to 1/n?

• 1/100 = 0.01
• 1/1000 = 0.001
• 1/1000000 = 0.000001

It gets closer and closer to 0!

We say: The limit of 1/n as n approaches infinity is 0

Written as: lim(n→∞) 1/n = 0

📚 Series Definition: Adding It All Up

A series is what happens when you ADD all the terms of a sequence together.

Sequence: 1, 2, 3, 4, 5, … Series: 1 + 2 + 3 + 4 + 5 + …

We use this fancy symbol: Σ (Sigma, the Greek letter for “Sum”)

∞
Σ aₙ = a₁ + a₂ + a₃ + a₄ + ...
n=1

Simple Example:

Series: 1 + 2 + 3 + 4 + 5

This equals 15. Easy!

But what if we add FOREVER? That’s an infinite series.

🧮 Partial Sums: Baby Steps to Infinity

We can’t add infinite numbers at once. So we use partial sums!

A partial sum is just: “How much do we have after adding the first n terms?”

Example:

Series: 1 + 1/2 + 1/4 + 1/8 + …

See the pattern? We’re getting closer and closer to 2!

graph TD
    A["S₁ = 1"] --> B["S₂ = 1.5"]
    B --> C["S₃ = 1.75"]
    C --> D["S₄ = 1.875"]
    D --> E["S₅ = 1.9375"]
    E --> F["... approaches 2"]

⚖️ Convergence vs Divergence: Does It Land or Fly Away?

This is the BIG question for every infinite series!

Convergent Series ✅

If the partial sums get closer and closer to a specific number, the series converges.

Example: 1 + 1/2 + 1/4 + 1/8 + … = 2

Like our shrinking steps - you approach a destination!

Divergent Series ❌

If the partial sums keep growing forever (or bouncing around), the series diverges.

Example: 1 + 2 + 3 + 4 + 5 + … = ∞

Like trying to stack unlimited full-sized coins - it never stops growing!

Quick Test:

If the terms don’t shrink to zero, the series DEFINITELY diverges!

But wait - terms shrinking to zero doesn’t guarantee convergence. We need more tools!

🔷 Geometric Series: The Multiplying Pattern

A geometric series has a special pattern: each term is the previous term times the same number (called r, the ratio).

Formula:

a + ar + ar² + ar³ + ar⁴ + ...

• a = first term
• r = common ratio

The Magic Rule:

Example 1: r = 1/2 (Converges!)

1 + 1/2 + 1/4 + 1/8 + ...

• a = 1
• r = 1/2

Sum = 1/(1 - 1/2) = 1/(1/2) = 2 ✅

Example 2: r = 2 (Diverges!)

1 + 2 + 4 + 8 + 16 + ...

Each term DOUBLES! This explodes to infinity. ❌

🎵 Harmonic Series: The Surprising Troublemaker

The harmonic series looks innocent:

1 + 1/2 + 1/3 + 1/4 + 1/5 + ...

Each term gets smaller, heading toward zero. Surely it converges?

PLOT TWIST: It Diverges! 😱

Even though the terms shrink, they don’t shrink FAST enough!

Here’s the proof trick (grouping terms):

1 + 1/2 + (1/3 + 1/4) + (1/5+1/6+1/7+1/8) + ...
    1/2 +   (>1/2)    +     (>1/2)        + ...

We keep adding amounts greater than 1/2, forever!

Lesson: Terms going to zero is NOT enough. Speed matters!

📊 P-Series Test: The Power Check

A p-series looks like this:

1/1ᵖ + 1/2ᵖ + 1/3ᵖ + 1/4ᵖ + ...

The p is the power on the bottom.

The Simple Rule:

Examples:

p = 2 (Converges!):

1 + 1/4 + 1/9 + 1/16 + ... = π²/6 ≈ 1.645

The terms shrink fast enough!

p = 1 (Diverges!):

1 + 1/2 + 1/3 + 1/4 + ...

This is the harmonic series - we just learned it diverges!

p = 1/2 (Diverges!):

1 + 1/√2 + 1/√3 + 1/√4 + ...

Terms don’t shrink fast enough.

🗺️ The Complete Picture

graph TD
    A["Infinite Series"] --> B{Do terms → 0?}
    B -->|No| C["DIVERGES ❌"]
    B -->|Yes| D{What type?}
    D --> E["Geometric: check r"]
    D --> F["P-series: check p"]
    D --> G["Other tests..."]
    E -->|r < 1| H["CONVERGES ✅"]
    E -->|r ≥ 1| C
    F -->|p > 1| H
    F -->|p ≤ 1| C

🎁 Quick Recap

💡 The Big Takeaway

Infinite series are like a race between adding and shrinking.

• If terms shrink FAST enough → Converges (you reach a destination)
• If terms shrink TOO slowly → Diverges (you wander forever)

The geometric series and p-series tests are your first tools to figure out who wins the race!

Now you’re ready to explore the infinite! 🚀