Series

🎯 Series: The Magic of Adding Things Up!

The Story of Adding Together

Imagine you have a piggy bank. Every week, you drop in some coins. A sequence is like looking at each coin one by one. But a series? That’s when you dump out all the coins and count the total!

Series = Adding up all the terms in a sequence

Let’s discover the magical shortcuts to count your treasure without counting every single coin!

🔢 Sigma Notation: The Superhero Symbol

Meet Σ (called “Sigma”) — the superhero symbol that means “add everything up!”

What Does It Look Like?

  5
  Σ  n  =  1 + 2 + 3 + 4 + 5  =  15
 n=1

Breaking It Down (Like a Recipe!)

Simple Example

“Add the first 4 even numbers”

  4
  Σ  2k  =  2(1) + 2(2) + 2(3) + 2(4)
 k=1
        =  2 + 4 + 6 + 8
        =  20

💡 Think of it like this: Sigma is a robot that follows instructions. You tell it where to start, where to stop, and what to add!

➕ Arithmetic Series: Adding Equal Steps

The Piggy Bank with Equal Deposits

Imagine you save money like this:

• Week 1: $5
• Week 2: $8
• Week 3: $11
• Week 4: $14

You’re adding $3 more each week! That’s an arithmetic sequence.

An arithmetic series is the total of all these amounts.

The Magic Formula

Instead of adding one by one, use this shortcut:

        n
  Sₙ = — × (first term + last term)
        2

Or written another way:

        n
  Sₙ = — × (a₁ + aₙ)
        2

Real Example

Add: 5 + 8 + 11 + 14

• n (how many terms) = 4
• First term (a₁) = 5
• Last term (aₙ) = 14

      4
Sₙ = — × (5 + 14)
      2

   = 2 × 19
   = 38

Check: 5 + 8 + 11 + 14 = 38 ✓

Another Handy Formula

When you know the first term and the common difference (d):

        n
  Sₙ = — × [2a₁ + (n-1)d]
        2

Example: Find the sum of first 10 terms where a₁ = 3 and d = 4

       10
  S₁₀ = — × [2(3) + (10-1)(4)]
        2

     = 5 × [6 + 36]
     = 5 × 42
     = 210

✖️ Geometric Series: Multiplying Magic

The Folding Paper Story

Fold a paper in half. You have 2 layers. Fold again. Now 4 layers. Again. Now 8 layers!

Each time you multiply by 2. That’s a geometric sequence.

Adding these up: 2 + 4 + 8 + … is a geometric series!

The Multiplication Pattern

The Magic Formula

        a₁ × (1 - rⁿ)
  Sₙ = ———————————————   (when r ≠ 1)
            1 - r

Where:

• a₁ = first term
• r = what you multiply by (common ratio)
• n = how many terms

Real Example

Find: 3 + 6 + 12 + 24 + 48

• a₁ = 3
• r = 2 (each term is 2× the previous)
• n = 5 terms

       3 × (1 - 2⁵)
  S₅ = ——————————————
           1 - 2

       3 × (1 - 32)
     = ——————————————
           -1

       3 × (-31)
     = ——————————
          -1

     = 93

Check: 3 + 6 + 12 + 24 + 48 = 93 ✓

📐 Sum Formulas: Your Cheat Sheet

The Big Three Formulas

graph TD
    A["Sum Formulas"] --> B["Arithmetic"]
    A --> C["Geometric"]
    A --> D["Special"]
    B --> E["Sₙ = n/2 × #40;a₁ + aₙ#41;"]
    C --> F["Sₙ = a₁#40;1-rⁿ#41;/#40;1-r#41;"]
    D --> G["1+2+...+n = n#40;n+1#41;/2"]

Special Sum: Adding 1 to n

Want to add 1 + 2 + 3 + … + n? There’s a legendary shortcut!

The Story of Young Gauss: A teacher told students to add 1 to 100. Young Gauss figured it out in seconds:

1 + 100 = 101
2 + 99  = 101
3 + 98  = 101
...

There are 50 pairs, each adding to 101!

Sum = 50 × 101 = 5050

The Formula

                n × (n + 1)
  1 + 2 + ... + n = ———————————
                      2

Example: Add 1 + 2 + 3 + … + 20

      20 × 21
  S = ————————
         2

    = 420 ÷ 2
    = 210

♾️ Infinite Geometric Series: Forever Adding!

The Cookie Story

Imagine you eat half a cookie. 🍪 Then half of what’s left (¼). Then half of that (⅛). Then half of that (1/16)…

Will you ever eat the whole cookie?

Amazingly… YES! As you keep going forever, you get closer and closer to eating exactly 1 whole cookie!

When Can We Add Forever?

Only when |r| < 1 (the common ratio is between -1 and 1)

This means each term gets smaller and smaller, heading toward zero.

The Infinite Sum Formula

        a₁
  S∞ = ——————   (only when |r| < 1)
       1 - r

Real Example 1

Find: 1/2 + 1/4 + 1/8 + 1/16 + … (forever!)

• a₁ = 1/2
• r = 1/2 (each term is half the previous)

       1/2
  S∞ = ————————
       1 - 1/2

       1/2
     = ————
       1/2

     = 1

The infinite sum equals exactly 1! 🎉

Real Example 2

Find: 8 + 4 + 2 + 1 + 0.5 + … (forever!)

• a₁ = 8
• r = 1/2

        8
  S∞ = ————————
       1 - 1/2

        8
     = ————
       0.5

     = 16

When It DOESN’T Work

If |r| ≥ 1, the sum explodes to infinity!

Example: 2 + 4 + 8 + 16 + … (r = 2)

This keeps getting bigger forever — no finite answer exists!

🎯 Quick Decision Guide

graph TD
    A["Is it a series?"] -->|Adding terms| B{What type?}
    B -->|Same difference| C["Arithmetic"]
    B -->|Same ratio| D["Geometric"]
    C --> E["Use: n/2#40;a₁+aₙ#41;"]
    D --> F{Finite or Infinite?}
    F -->|Finite n terms| G["Use: a₁#40;1-rⁿ#41;/#40;1-r#41;"]
    F -->|"Infinite & │r│<1"| H["Use: a₁/&#35;40;1-r&#35;41;"]

🌟 Summary: What We Learned

💪 You’ve Got This!

Series are just smart ways to add things up without doing all the work. Whether you’re:

• Calculating interest in a bank 🏦
• Figuring out total distance traveled 🚗
• Computing probabilities in games 🎮

…these formulas are your superpowers!

Remember: A journey of a thousand sums begins with understanding Σ (sigma)! 🚀