Calculus 2 Sequences And Series Cheat Sheet

Calculus 2 Sequences and Series Cheat Sheet

Navigating the world of sequences and series in Calculus 2 can feel like learning a new language. It’s a shift from the concrete integrals and derivatives of Calculus 1 to the more abstract, proof-oriented analysis of infinite processes. This Calculus 2 sequences and series cheat sheet is designed to be your concise, yet comprehensive, field guide. It distills the essential definitions, theorems, tests, and formulas into a structured reference, clarifying not just what each concept is, but when and why to use it. Mastery here isn't about memorizing every test; it's about developing a strategic intuition for diagnosing the behavior of an infinite sum.

1. The Foundation: Sequences

A sequence is simply an ordered list of numbers, typically defined by a formula (a_n). The fundamental question is: what happens to (a_n) as (n) approaches infinity?

• Limit of a Sequence: (\lim_{n \to \infty} a_n = L). The sequence converges to (L) if the terms get arbitrarily close to (L) and stay there. If the limit does not exist (including diverging to (\pm\infty)), the sequence diverges.
• Key Tools for Limits:
  • Continuous Function Theorem: If (\lim_{n \to \infty} b_n = L) and (f) is continuous at (L), then (\lim_{n \to \infty} f(b_n) = f(L)). Useful for limits involving roots, logs, etc.
  • Squeeze (Sandwich) Theorem: If (b_n \leq a_n \leq c_n) for all (n) beyond some point, and (\lim b_n = \lim c_n = L), then (\lim a_n = L). Your go-to for sequences with oscillating terms like (\frac{\sin n}{n}).
  • L'Hôpital's Rule: Applicable for indeterminate forms (\frac{\infty}{\infty}) or (\frac{0}{0}) when treating (n) as a continuous variable (x). Use cautiously; it's often overkill for simple polynomial/radical sequences.

2. The Core Objective: Infinite Series

An infinite series is the sum of the terms of an infinite sequence: (\sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \dots). The core concept is the sequence of partial sums: (S_N = \sum_{n=1}^{N} a_n). The series converges if the sequence ({S_N}) converges to a finite limit (S). If ({S_N}) diverges, the series diverges.

Crucial First Step: The nth-Term Test for Divergence

This is your immediate diagnostic tool. If (\lim_{n \to \infty} a_n \neq 0), then (\sum a_n) must diverge. Warning: If (\lim_{n \to \infty} a_n = 0), the test is inconclusive. The series may converge (e.g., (\sum \frac{1}{n^2})) or diverge (e.g., (\sum \frac{1}{n})).

3. Convergence Tests: Your Strategic Toolkit

Think of these tests as different keys for different locks. No single test works for everything. Your strategy is to examine the general term (a_n) and choose the most promising test.

For Series with Positive Terms ((a_n > 0))

This is the most common and simplest category.

1. Geometric Series: (\sum ar^n).

   • Converges if (|r| < 1), to (\frac{a}{1-r}).
   • Diverges if (|r| \geq 1).
   • Why it matters: It's the only series for which we have a simple, exact sum formula. Many other series are compared to it.

2. p-Series: (\sum \frac{1}{n^p}).

   • Converges if (p > 1).
   • Diverges if (p \leq 1).
   • Why it matters: It's a fundamental benchmark. The harmonic series ((p=1)) is the classic divergent example.

3. Integral Test: Applies if (a_n = f(n)), where (f) is continuous, positive, and decreasing for (x \geq N).

   • (\sum a_n) and (\int_{N}^{\infty} f(x) dx) both converge or both diverge.
   • Use when: The function (f(x)) is easily integrable. The test gives convergence/divergence but not the sum.

4. Comparison Tests (Direct & Limit):

   • Direct Comparison Test: If (0 \leq a_n \leq b_n) and (\sum b_n) converges, then (\sum a_n) converges. If (0 \leq b_n \leq a_n) and (\sum b_n) diverges, then (\sum a_n) diverges. Requires finding a suitable "benchmark" series.
   • Limit Comparison Test: For (a_n, b_n > 0), compute (L = \lim_{n \to \infty

} \frac{a_n}{b_n}). * If (0 < L < \infty), then (\sum a_n) and (\sum b_n) both converge or both diverge. * Use when: The terms (a_n) are a rational function or a combination of powers and roots. It's more flexible than direct comparison.

5. Ratio Test: Compute (L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|).

   • If (L < 1), the series converges absolutely.
   • If (L > 1) or (L = \infty), the series diverges.
   • If (L = 1), the test is inconclusive.
   • Use when: The terms involve factorials ((n!)) or exponentials ((r^n)). It's the primary test for power series.

6. Root Test: Compute (L = \lim_{n \to \infty} \sqrt[n]{|a_n|}).

   • If (L < 1), the series converges absolutely.
   • If (L > 1) or (L = \infty), the series diverges.
   • If (L = 1), the test is inconclusive.
   • Use when: The general term is of the form (a_n = (b_n)^n). It's less commonly used than the ratio test but can be powerful in specific cases.

For Series with Mixed Signs

1. Alternating Series Test: For (\sum (-1)^n b_n) or (\sum (-1)^{n+1} b_n) where (b_n > 0).

   • If (b_{n+1} \leq b_n) for all (n) (eventually) and (\lim_{n \to \infty} b_n = 0), then the series converges.
   • Use when: The series is alternating. It's the only test specifically for alternating series.

2. Absolute Convergence: A series (\sum a_n) converges absolutely if (\sum |a_n|) converges.

   • If a series converges absolutely, it converges (regardless of sign changes).
   • Strategy: If a series has mixed signs, first check if (\sum |a_n|) converges (using tests for positive terms). If it does, you're done. If not, the series might still converge conditionally (e.g., an alternating series that passes the alternating series test but fails the absolute convergence test).

4. A Strategic Approach: Choosing the Right Test

There is no universal test. Your strategy is to analyze the form of (a_n):

1. First, apply the nth-term test. If (\lim_{n \to \infty} a_n \neq 0), the series diverges. Stop.
2. Identify the series type:
   • Is it a geometric series ((\sum ar^n))? Use the geometric series test.
   • Is it a p-series ((\sum \frac{1}{n^p}))? Use the p-series rule.
   • Is it alternating ((\sum (-1)^n b_n))? Use the alternating series test.
3. Look at the structure of (a_n):
   • Does it involve (n!) or (r^n)? Try the ratio test.
   • Is it of the form ((b_n)^n)? Try the root test.
   • Is it a rational function or a combination of powers and roots? Try the limit comparison test with a p-series.
   • Can you write it as (f(n)) where (f(x)) is easily integrable? Try the integral test.
4. If all else fails, try the direct comparison test by finding a suitable benchmark series.

Warning: The ratio and root tests are inconclusive when (L=1). The comparison tests require careful selection of a benchmark series. The integral test requires the function to be decreasing.

5. The Bigger Picture: Why This Matters

Mastering infinite sequences and series is not about memorizing a list of tests. It's about developing a problem-solving mindset. You learn to:

• Analyze the structure of a problem.
• Choose the right tool for the job.
• Understand the limitations of each tool.
• Build logical arguments step-by-step.

This is the same mindset required for advanced calculus, differential equations, and beyond. The ability to dissect a complex sum and determine its fate—convergence or divergence—is a fundamental skill that underpins much of higher mathematics. It's the difference between seeing a series as an impenetrable wall of symbols and seeing it as a puzzle with a solution.