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5.4 Comparison Tests

3 min read•june 24, 2024

Comparison tests are powerful tools for determining if a converges or diverges. By comparing a tricky series to one with known behavior, we can draw conclusions about without directly analyzing the original series.

These tests come in two flavors: direct comparison and limit comparison. Each has its strengths, with direct comparison relying on inequalities and limit comparison using ratios of terms. Choosing the right comparison series is key to success.

Comparison Tests for Series Convergence

Application of comparison test

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  • Determines convergence or by comparing it to a known convergent or divergent series (, )
  • Ensures terms of the comparison series are greater than or equal to corresponding terms of the given series for all [n](https://www.fiveableKeyTerm:n)[n](https://www.fiveableKeyTerm:n) greater than some NN
  • Concludes the given series converges if the comparison series converges
  • Concludes the given series diverges if the comparison series diverges
  • Useful when the given series has terms that are difficult to work with directly but can be bounded by a simpler series (series with factorials, exponentials, or powers)

Limit comparison test analysis

  • Applies when direct comparison of series terms is not possible or convenient
  • Chooses a series with known convergence behavior similar in structure to the given series
  • Calculates the limit of the of corresponding terms of the two series as nn approaches infinity: lim⁡n→∞anbn\lim_{n\to\infty} \frac{a_n}{b_n}, where ana_n are terms of the given series and bnb_n are terms of the comparison series
  • Concludes both series either converge or diverge if the limit is a positive, finite value or infinity
  • Remains inconclusive if the limit is zero, requiring other methods to determine convergence
  • Particularly useful when the given series has terms that can be simplified using

Strategic comparison series choices

  • Selects appropriate comparison series crucial for effectively applying comparison tests
  • Considers common comparison series:
    1. p-series: ∑n=1∞1np\sum_{n=1}^{\infty} \frac{1}{n^p}
      • Converges for p>1p > 1 and diverges for p≤1p \leq 1
      • Compares series with polynomial terms in the denominator
    2. Geometric series: ∑n=0∞arn\sum_{n=0}^{\infty} ar^n
      • Converges for ∣r∣<1|r| < 1 and diverges for ∣r∣≥1|r| \geq 1
      • Compares series with exponential terms or constant ratios between successive terms
  • Analyzes the dominant term or behavior of the given series
    • For series with polynomial terms, considers a p-series with the same power
    • For series with exponential terms, considers a geometric series with the same base
  • Adjusts coefficients or constants in the comparison series to ensure the comparison is valid, with comparison series terms greater than or equal to the given series terms

Bounding and Inequalities in Comparison Tests

  • Uses inequalities to establish upper and lower bounds for series terms
  • Compares the ratio of corresponding terms between series to determine convergence behavior
  • Applies upper bounds to show convergence by comparing with a known convergent series
  • Utilizes lower bounds to demonstrate by comparing with a known divergent series

Key Terms to Review (20)

Alternating series: An alternating series is a series whose terms alternate in sign. It can be expressed as $\sum (-1)^n a_n$ or $\sum (-1)^{n+1} a_n$, where $a_n$ is a sequence of positive terms.
An: The term 'an' is a type of indefinite article in the English language, used to indicate a singular, non-specific noun. It is typically used before words that begin with a vowel sound, and it serves to introduce a new or unspecified entity.
Bn: The term 'bn' refers to the nth partial sum of an infinite series. It represents the sum of the first n terms of the series, which is used in the context of convergence and divergence tests, such as the Comparison Test covered in Section 5.4.
Comparison test: The Comparison Test is used to determine the convergence or divergence of an infinite series by comparing it to another series with known behavior. It involves either the Direct Comparison Test or the Limit Comparison Test.
Comparison Test: The comparison test is a method used to determine the convergence or divergence of a series or improper integral by comparing it to another series or integral with known convergence properties. It is a powerful tool for analyzing the behavior of infinite series and integrals.
Convergence: Convergence is a fundamental concept in mathematics that describes the behavior of sequences, series, and functions as they approach a specific value or limit. It is a crucial idea that underpins many areas of calculus, including the definite integral, improper integrals, direction fields, numerical methods, sequences, infinite series, and power series.
Divergence: Divergence is a fundamental concept in mathematics that describes the behavior of a sequence, series, or function as it approaches or departs from a specific value or pattern. This term is particularly relevant in the context of improper integrals, sequences, infinite series, comparison tests, ratio and root tests, and power series and functions.
Divergence of a series: The divergence of a series occurs when the sum of its terms does not approach a finite limit as more terms are added. A divergent series either increases without bound, decreases without bound, or oscillates indefinitely.
Geometric Series: A geometric series is an infinite series where each term is a constant multiple of the previous term. It is a type of infinite series that follows a specific pattern, allowing for the calculation of the sum of the series under certain conditions.
Inequality: Inequality refers to the state of being unequal or the lack of equality, particularly in the distribution or balance of something. In the context of calculus, inequality is a fundamental concept that describes the comparative relationship between two or more mathematical expressions or quantities.
L'Hôpital's Rule: L'Hôpital's rule is a powerful technique used to evaluate limits of indeterminate forms, such as $0/0$ or $\infty/\infty$. It states that if the limit of a ratio of functions is an indeterminate form, then the limit can be found by taking the ratio of the derivatives of the numerator and denominator functions.
Limit comparison test: The limit comparison test is a method to determine the convergence or divergence of an infinite series by comparing it to another series with known behavior. It involves taking the limit of the ratio of terms from two different series.
Limit Comparison Test: The limit comparison test is a method used to determine the convergence or divergence of a series by comparing it to another series with known convergence or divergence properties. It is a powerful tool for analyzing the behavior of infinite series and sequences.
Lower Bound: A lower bound is a value that sets the minimum or lowest possible limit for a mathematical quantity or function. It represents the smallest value that a variable or expression can take on within a given context or set of constraints.
N: The variable 'n' is a commonly used mathematical symbol that represents an arbitrary positive integer or a placeholder for a specific natural number. It is frequently employed in the context of series, sequences, and mathematical expressions to denote a general or unspecified quantity.
P-series: A p-series is a type of infinite series where the general term of the series is given by $\frac{1}{n^p}$, where $p$ is a real number. The convergence or divergence of a p-series is determined by the value of $p$, which is a crucial concept in the context of the Divergence and Integral Tests, Comparison Tests, and Ratio and Root Tests.
Ratio: A ratio is a quantitative relationship between two or more values, quantities, or magnitudes, typically expressed as a fraction or a quotient. Ratios are used to compare and analyze relative proportions or sizes within a given context.
Series: A series is a special type of sequence where each term in the sequence is added together to form a sum. Series are often used to model and analyze various mathematical phenomena, particularly in the context of convergence and divergence.
Summation Notation (∑): The summation notation, represented by the Greek letter Sigma (∑), is a concise way to express the sum of a series of terms or values. It is a fundamental concept in mathematics, particularly in the context of series, sequences, and various calculus topics.
Upper Bound: An upper bound is a value that is greater than or equal to all the elements in a set. It represents the maximum possible value or limit that a variable or function can attain within a given context.
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Glossary
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5.5 Alternating Series