Convergence Tests

Infinite Series: The Convergence Detective’s Toolkit 🔍

Imagine you’re a detective, and every infinite series is a mystery. Does it add up to a real number (converge), or does it fly off to infinity (diverge)? You have SIX powerful tools in your detective kit. Let’s master each one!

The Big Picture: What Are We Solving?

Think of an infinite series like stacking blocks forever: $\sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + a_4 + \ldots$

The Question: If we keep adding blocks forever, do we get a stable tower (converges) or does it topple over into infinity (diverges)?

graph TD
    A["Infinite Series"] --> B{Does it converge?}
    B -->|Yes| C["Adds up to a finite number"]
    B -->|No| D["Goes to infinity or oscillates"]
    B -->|Not sure| E["Use our 6 tests!"]

Test 1: The Divergence Test 🚦

The Simplest Check First!

The Rule: If the terms don’t shrink to zero, the series MUST diverge.

Think of it like this: If you’re trying to fill a bucket but each scoop of water is getting BIGGER, you’ll overflow. The scoops MUST get smaller to have any hope of stopping.

How It Works

$\text{If } \lim_{n \to \infty} a_n \neq 0, \text{ then } \sum a_n \text{ diverges}$

Example Time! 🎯

Series: $\sum_{n=1}^{\infty} \frac{n}{n+1}$

Let’s check: What happens to each term as $n$ gets huge?

$\lim_{n \to \infty} \frac{n}{n+1} = \lim_{n \to \infty} \frac{1}{1 + 1/n} = 1$

The terms approach 1, not 0!

Verdict: DIVERGES!

⚠️ The Trap!

If the limit IS zero, you can’t conclude anything! This test only catches obvious failures. It’s like checking if someone has a pulse—being alive doesn’t mean they’re healthy!

Test 2: The Integral Test 📊

Turn Sums Into Areas!

The Idea: Replace the discrete sum with a continuous integral. If the integral converges, so does the series!

Requirements (The Fine Print)

Your function $f(x)$ must be:

• Positive (stays above zero)
• Continuous (no breaks)
• Decreasing (always going down)

The Magic Connection

$\sum_{n=1}^{\infty} a_n \text{ and } \int_1^{\infty} f(x),dx \text{ behave the same way!}$

graph TD
    A["Series Terms"] --> B["Create function f where f n = aₙ"]
    B --> C{Check: Positive, Continuous, Decreasing?}
    C -->|Yes| D["Evaluate integral from 1 to ∞"]
    D --> E{Integral converges?}
    E -->|Yes| F["Series CONVERGES"]
    E -->|No| G["Series DIVERGES"]

Example Time! 🎯

Series: $\sum_{n=1}^{\infty} \frac{1}{n^2}$

Step 1: Let $f(x) = \frac{1}{x^2}$. This is positive, continuous, and decreasing for $x \geq 1$. ✓

Step 2: Evaluate the integral:

$\int_1^{\infty} \frac{1}{x^2},dx = \left[-\frac{1}{x}\right]_1^{\infty} = 0 - (-1) = 1$

The integral equals 1 (a finite number)!

Verdict: CONVERGES!

Test 3: The Comparison Test ⚖️

Find a Friend Series!

The Idea: Compare your mystery series to one you already know!

Two Scenarios

Scenario A: Your series is SMALLER than a converging series

• If the “big friend” converges, YOU converge too!

Scenario B: Your series is BIGGER than a diverging series

• If the “small friend” diverges, YOU diverge too!

graph TD
    A["Your Series aₙ"] --> B{Compare to known series bₙ}
    B --> C["If aₙ ≤ bₙ AND bₙ converges"]
    C --> D["aₙ CONVERGES"]
    B --> E["If aₙ ≥ bₙ AND bₙ diverges"]
    E --> F["aₙ DIVERGES"]

Example Time! 🎯

Series: $\sum_{n=1}^{\infty} \frac{1}{n^2 + 1}$

Find a friend: Notice that $\frac{1}{n^2 + 1} < \frac{1}{n^2}$

We know $\sum \frac{1}{n^2}$ converges (it’s a p-series with p=2 > 1).

Since our series is SMALLER than a converging series…

Verdict: CONVERGES!

Test 4: The Limit Comparison Test 🔬

Compare Using Ratios!

When to use: When regular comparison is tricky, but your series “looks like” a known series.

The Formula

$L = \lim_{n \to \infty} \frac{a_n}{b_n}$

If L is a positive finite number (not 0, not ∞): Both series do the SAME thing!

Example Time! 🎯

Series: $\sum_{n=1}^{\infty} \frac{2n+1}{n^3 - n}$

This looks like: $\frac{n}{n^3} = \frac{1}{n^2}$ for large n

Calculate the limit:

$L = \lim_{n \to \infty} \frac{\frac{2n+1}{n^3-n}}{\frac{1}{n^2}} = \lim_{n \to \infty} \frac{(2n+1)n^2}{n^3-n}$

$= \lim_{n \to \infty} \frac{2n^3 + n^2}{n^3 - n} = \lim_{n \to \infty} \frac{2 + 1/n}{1 - 1/n^2} = 2$

L = 2 is positive and finite!

Since $\sum \frac{1}{n^2}$ converges, so does our series!

Verdict: CONVERGES!

Test 5: The Ratio Test 📏

The Superstar of Series Tests!

The Idea: Compare each term to the NEXT term. If terms shrink fast enough, the series converges!

The Formula

$L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|$

Decode the Result:

• L < 1: CONVERGES (terms shrinking fast!)
• L > 1: DIVERGES (terms growing!)
• L = 1: INCONCLUSIVE (need different test)

graph TD
    A["Calculate L = limit of |aₙ₊₁/aₙ|"] --> B{What is L?}
    B -->|L < 1| C["CONVERGES absolutely"]
    B -->|L > 1| D["DIVERGES"]
    B -->|L = 1| E["Test is INCONCLUSIVE"]

Example Time! 🎯

Series: $\sum_{n=1}^{\infty} \frac{n!}{3^n}$

Calculate the ratio:

$\frac{a_{n+1}}{a_n} = \frac{(n+1)!/3^{n+1}}{n!/3^n} = \frac{(n+1)! \cdot 3^n}{n! \cdot 3^{n+1}} = \frac{n+1}{3}$

Take the limit:

$L = \lim_{n \to \infty} \frac{n+1}{3} = \infty$

L = ∞, which is definitely > 1!

Verdict: DIVERGES!

💡 Pro Tip!

The Ratio Test is AMAZING for series with:

• Factorials (n!)
• Exponentials (2ⁿ, 3ⁿ)
• Products of both

Test 6: The Root Test 🌱

The Ratio Test’s Cousin!

The Idea: Take the nth root of the nth term. Perfect for terms with nth powers!

The Formula

$L = \lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} |a_n|^{1/n}$

Same Rules as Ratio Test:

• L < 1: CONVERGES
• L > 1: DIVERGES
• L = 1: INCONCLUSIVE

Example Time! 🎯

Series: $\sum_{n=1}^{\infty} \left(\frac{2n+1}{3n+2}\right)^n$

Perfect for Root Test! The term is already an nth power!

$\sqrt[n]{|a_n|} = \sqrt[n]{\left(\frac{2n+1}{3n+2}\right)^n} = \frac{2n+1}{3n+2}$

Take the limit:

$L = \lim_{n \to \infty} \frac{2n+1}{3n+2} = \lim_{n \to \infty} \frac{2 + 1/n}{3 + 2/n} = \frac{2}{3}$

L = 2/3, which is < 1!

Verdict: CONVERGES!

💡 When to Use Root vs Ratio?

The Detective’s Decision Tree 🌳

When facing a new series, follow this path:

graph TD
    A["New Series"] --> B{Does aₙ → 0?}
    B -->|No| C["DIVERGES - Divergence Test"]
    B -->|Yes| D{Looks like known series?}
    D -->|Yes| E["Try Comparison or Limit Comparison"]
    D -->|No| F{Has factorials or exponentials?}
    F -->|Yes| G["Try Ratio Test"]
    F -->|No| H{Has nth powers?}
    H -->|Yes| I["Try Root Test"]
    H -->|No| J{Is f nice positive decreasing?}
    J -->|Yes| K["Try Integral Test"]
    J -->|No| L["Keep experimenting!"]

Quick Reference: The Six Heroes

You’ve Got This! 🎉

Remember:

1. Always check Divergence Test first - it’s free and fast!
2. Look for patterns - factorials scream “Ratio Test!”
3. Compare to friends - p-series and geometric series are your buddies
4. Practice makes perfect - each test becomes second nature!

You’re now a Convergence Detective with a full toolkit. Go solve those infinite mysteries! 🔍✨