Sequences & Series

TL;DR

Sequences are ordered lists of numbers that follow a pattern, while series are the sums of the terms in a sequence. Understanding their convergence or divergence is crucial for many calculus applications, like Taylor series. We use tests to determine if an infinite series adds up to a finite number.

1. The Mental Model

Think of a sequence like steps on a staircase: each step is a number in a specific order. A series is like the total height you've climbed after taking a certain number of steps. We're often interested in whether the staircase eventually reaches a finite height (converges) or just keeps going up forever (diverges).

2. The Core Material

What's a Sequence?

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A sequence is just an ordered list of numbers, often defined by a formula for its $n$-th term, written as $\{a_n\}$.
For example, the sequence $\{1/n\}$ means:
$a_1 = 1/1 = 1$
$a_2 = 1/2$
$a_3 = 1/3$
...and so on.

A sequence converges if its terms approach a single finite number as $n$ gets really big (as $n \to \infty$). Otherwise, it diverges. For $\{1/n\}$, as $n \to \infty$, $1/n \to 0$, so it converges to 0.

What's a Series?

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A series is the sum of the terms of a sequence. It's written as $\sum_{n=1}^{\infty} a_n$.
Using our example sequence $\{1/n\}$, the corresponding series is $\sum_{n=1}^{\infty} 1/n = 1 + 1/2 + 1/3 + 1/4 + \dots$.

A series converges if the sum of its terms approaches a single finite number. If the sum keeps growing without bound, or oscillates, it diverges.

Convergence Tests for Series

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It's usually impossible to sum infinitely many terms directly. So, we have tests:

1. Divergence Test (or $n$-th Term Test)

If $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum a_n$ diverges.
Crucial: If $\lim_{n \to \infty} a_n = 0$, this test tells you nothing about convergence. The series might converge or diverge.

2. Geometric Series Test

A geometric series looks like $\sum_{n=0}^{\infty} ar^n = a + ar + ar^2 + \dots$.
It converges if $|r| < 1$, and its sum is $a/(1-r)$.
It diverges if $|r| \ge 1$.

3. p-Series Test

A p-series looks like $\sum_{n=1}^{\infty} 1/n^p$.
It converges if $p > 1$.
It diverges if $p \le 1$. (The harmonic series $\sum 1/n$ is a p-series with $p=1$, and it diverges).

4. Integral Test (Conditions Apply!)

If $f(x)$ is positive, continuous, and decreasing for $x \ge 1$, and $a_n = f(n)$, then $\sum_{n=1}^{\infty} a_n$ and $\int_{1}^{\infty} f(x) \, dx$ either both converge or both diverge.

5. Comparison Tests

These help you compare a difficult series to one you already know (like a p-series or geometric series).

• Direct Comparison Test: If $0 \le a_n \le b_n$ for all $n$ large enough:
  • If $\sum b_n$ converges, then $\sum a_n$ converges.
  • If $\sum a_n$ diverges, then $\sum b_n$ diverges.
• Limit Comparison Test (LCT): If $a_n > 0$ and $b_n > 0$, and $\lim_{n \to \infty} (a_n/b_n) = L$ where $L$ is a finite positive number ($0 < L < \infty$), then both $\sum a_n$ and $\sum b_n$ either converge or both diverge.

6. Alternating Series Test (Leibniz Test)

For an alternating series $\sum_{n=1}^{\infty} (-1)^{n-1} b_n$ (where $b_n > 0$):
It converges if:
1. $\lim_{n \to \infty} b_n = 0$
2. $b_n$ is a decreasing sequence (i.e., $b_{n+1} \le b_n$ for all $n$)

7. Ratio Test

This is great for series with factorials or powers of $n$.
Let $L = \lim_{n \to \infty} |a_{n+1}/a_n|$.
* If $L < 1$, the series converges absolutely.
* If $L > 1$, the series diverges.
* If $L = 1$, the test is inconclusive (use another test).

graph TD
    A["Start: Consider a Series \(\sum a_n\)"] --> B{"Is it Geometric \(\sum ar^n\)"};
    B -- Yes --> C{"Is |r| < 1?"};
    C -- Yes --> D["Converges (sum a/(1-r))"];
    C -- No --> E["Diverges"];
    B -- No --> F{"Is it a p-Series \(\sum 1/n^p\)"};
    F -- Yes --> G{"Is p > 1?"};
    G -- Yes --> D;
    G -- No --> E;
    F -- No --> H{"Does \(\lim_{n \to \infty} a_n \neq 0\)"};
    H -- Yes (Divergence Test) --> E;
    H -- No (Test is Inconclusive) --> I{"Does it involve factorials or powers of n?"};
    I -- Yes --> J{"Try Ratio Test"};
    J --> K{"Is L < 1?"};
    K -- Yes --> D;
    K -- No, is L > 1? --> E;
    K -- No, L=1? --> L["Ratio Test Inconclusive, Try Another"];
    I -- No --> M{"Is it an Alternating Series?"};
    M -- Yes --> N{"Apply Alternating Series Test"};
    N -- Both conditions met? --> D;
    N -- No --> L;
    M -- No --> O{"Try Comparison Tests (Direct or Limit)"};
    O --> P{"Can you compare it to a known series?"};
    P -- Yes --> D_or_E_via_Comparison["Converges/Diverges (based on comparison)"];
    P -- No --> Q{"Try Integral Test (if conditions met)"};
    Q --> D_or_E_via_Integral["Converges/Diverges (based on integral)"];
    Q -- No --> R["Series is Tricky! May need advanced methods."];

3. Worked Example

Let's determine if the series $\sum_{n=1}^{\infty} \frac{n}{e^n}$ converges or diverges.

1. Divergence Test: $\lim_{n \to \infty} \frac{n}{e^n}$. Using L'Hôpital's Rule: $\lim_{n \to \infty} \frac{1}{e^n} = 0$. The test is inconclusive.

2. Ratio Test: This often works well with $e^n$.
   $a_n = \frac{n}{e^n}$
   $a_{n+1} = \frac{n+1}{e^{n+1}}$

   $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{(n+1)/e^{n+1}}{n/e^n} \right|$
   $L = \lim_{n \to \infty} \left| \frac{n+1}{e^{n+1}} \cdot \frac{e^n}{n} \right|$
   $L = \lim_{n \to \infty} \left| \frac{n+1}{n} \cdot \frac{e^n}{e \cdot e^n} \right|$
   $L = \lim_{n \to \infty} \left| \left(1 + \frac{1}{n}\right) \cdot \frac{1}{e} \right|$
   $L = \left(1 + 0\right) \cdot \frac{1}{e} = \frac{1}{e}$

Since $L = 1/e \approx 1/2.718 < 1$, the series converges by the Ratio Test.

4. Key Takeaways

• A sequence is an ordered list, while a series is the sum of a sequence's terms.
• For a series to converge, its terms must eventually approach zero (Divergence Test), but this alone isn't enough.
• Geometric series converge if $|r|<1$, and p-series converge if $p>1$.
• The Ratio Test is powerful for series with factorials or exponents.
• Comparison tests allow you to deduce convergence/divergence by relating a series to a known one.
• The Alternating Series Test has specific conditions for alternating series to converge.

Common Mistakes to Avoid:
- Assuming a series converges just because its terms go to zero (e.g., the harmonic series).
- Mixing up the conditions for different tests, especially for comparison tests.
- Forgetting that the Integral Test, Comparison Tests, and Ratio Test typically require positive terms.
- Incorrectly calculating limits, especially when using L'Hôpital's Rule.

5. Now Try It

Determine if the series $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$ converges absolutely, converges conditionally, or diverges. Write down which test(s) you used and your reasoning.

What success looks like: You correctly apply the Alternating Series Test for conditional convergence and then attempt to test for absolute convergence (using a p-series test or integral test on the absolute value of the terms) to classify the series accurately.