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5.3 The Divergence and Integral Tests

2 min read•june 24, 2024

The and are powerful tools for analyzing infinite . They help determine if a series converges or diverges, which is crucial for understanding its behavior and potential applications.

These tests, along with comparison methods and , form a toolkit for tackling various types of series. By mastering these techniques, you'll be able to evaluate complex sums and make informed decisions about their convergence or divergence.

The Divergence Test

Divergence test for series

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  • Determines if a series diverges by examining the of its terms
    • If lim⁡n→∞an≠0\lim_{n \to \infty} a_n \neq 0 or the limit does not exist, the series ∑n=1∞an\sum_{n=1}^{\infty} a_n diverges (∑n=1∞1n\sum_{n=1}^{\infty} \frac{1}{n}, )
    • If lim⁡n→∞an=0\lim_{n \to \infty} a_n = 0, the test is inconclusive and the series may converge or diverge (∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^2}, pp-series with p=2p=2)
  • Useful for identifying series quickly without calculating the sum
  • Does not provide information about convergence

The Integral Test

Integral test for convergence

  • Determines the convergence or divergence of a series by comparing it to an
  • For a series ∑n=1∞an\sum_{n=1}^{\infty} a_n, let f(x)f(x) be a continuous, positive, and decreasing function on [1,∞)[1, \infty) such that f(n)=anf(n) = a_n for all n≥1n \geq 1
    • If ∫1∞f(x)dx\int_1^{\infty} f(x) dx converges, then ∑n=1∞an\sum_{n=1}^{\infty} a_n converges (∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^2} and ∫1∞1x2dx\int_1^{\infty} \frac{1}{x^2} dx)
    • If ∫1∞f(x)dx\int_1^{\infty} f(x) dx diverges, then ∑n=1∞an\sum_{n=1}^{\infty} a_n diverges (∑n=1∞1n\sum_{n=1}^{\infty} \frac{1}{n} and ∫1∞1xdx\int_1^{\infty} \frac{1}{x} dx)
  • Useful for series involving logarithms, exponentials, or powers (∑n=1∞1nln⁡n\sum_{n=1}^{\infty} \frac{1}{n \ln n}, ∑n=1∞e−n\sum_{n=1}^{\infty} e^{-n})
  • Requires the function to be monotonic (decreasing) for the test to be valid

Remainder terms for series estimation

  • The RnR_n is the difference between the series value SS and the nn-th SnS_n
    • Rn=S−SnR_n = S - S_n, represents the error when approximating the series value with a
  • For a series ∑n=1∞an\sum_{n=1}^{\infty} a_n with positive, decreasing, and continuous terms ana_n for n≥1n \geq 1:
    1. ∫n+1∞andn≤Rn≤∫n∞andn\int_{n+1}^{\infty} a_n dn \leq R_n \leq \int_n^{\infty} a_n dn
    2. These integrals provide lower and upper bounds for RnR_n
  • Calculating remainder term bounds helps:
    • Estimate the series value (∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^2}, approximate value to within 0.010.01)
    • Determine the number of terms needed for a desired accuracy (∑n=1∞e−n\sum_{n=1}^{\infty} e^{-n}, how many terms for an error less than 0.0010.001?)

Additional Convergence Tests

Comparison and Limit Comparison Tests

  • Compares the given series to a known convergent or divergent series
  • If 0≤an≤bn0 \leq a_n \leq b_n for all n≥Nn \geq N:
    • If ∑bn\sum b_n converges, then ∑an\sum a_n converges
    • If ∑an\sum a_n diverges, then ∑bn\sum b_n diverges
  • Limit uses the limit of the ratio of terms
  • Useful for series that are difficult to evaluate directly

Telescoping Series

  • A series where terms cancel out, leaving only a finite number of terms
  • Often results in a simple expression for the sum
  • Example: ∑n=1∞(1n−1n+1)\sum_{n=1}^{\infty} (\frac{1}{n} - \frac{1}{n+1})

Key Terms to Review (26)

∞: Infinity, or the symbol ∞, represents a quantity without an end or limit. It is a concept that denotes something that is boundless, endless, or immeasurably large. In the context of calculus, infinity plays a crucial role in the understanding of convergence and divergence of series and integrals.
1/x^p: The term 1/x^p, where p is a positive real number, represents a specific type of function that is commonly encountered in the context of the Divergence Test and the Integral Test in calculus. This function belongs to the class of power functions and exhibits unique properties that are crucial for understanding the convergence or divergence of series.
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.
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.
Convergent: Convergent refers to a sequence or series that approaches a finite, specific value as the number of terms increases. In the context of calculus, convergence is a critical concept that determines the behavior and properties of infinite series and integrals.
Coupon collector’s problem: The coupon collector's problem is a classic problem in probability theory that determines the expected number of trials needed to collect all coupons from a set. It is often used to illustrate concepts of sequences and series in mathematical contexts.
Divergence test: The Divergence Test is a method used to determine whether a given series diverges. If the limit of the sequence's terms does not equal zero, the series diverges.
Divergence Test: The divergence test is a method used to determine the convergence or divergence of an infinite series by examining the behavior of the terms in the series. It is a fundamental tool in the study of series and their convergence properties.
Divergent: Divergent refers to a series of mathematical sequences or series that do not converge to a finite value. This term is particularly relevant in the context of the Divergence and Integral Tests, which are used to determine the convergence or divergence of infinite series.
Harmonic Series: The harmonic series is an infinite series where each term is the reciprocal of a positive integer. It is an important concept in the context of the Divergence and Integral Tests, as it serves as a model for understanding the behavior of infinite series.
Improper Integral: An improper integral is a type of integral that has a domain that extends to infinity or contains a point where the integrand is not defined. These integrals are used to study the convergence or divergence of infinite series and sequences.
Integral test: The integral test is a method to determine the convergence or divergence of an infinite series by comparing it to an improper integral. If the integral converges, so does the series, and if the integral diverges, so does the series.
Integral Test: The integral test is a method used to determine the convergence or divergence of an infinite series by comparing it to the integral of a related function. It provides a way to analyze the behavior of a series without having to explicitly evaluate each term.
Limit: A limit describes the value that a function approaches as the input approaches a certain point. This concept is fundamental in mathematics as it helps us understand behavior in calculus, particularly when dealing with continuity, derivatives, and integrals. Limits allow us to analyze the behavior of functions at points where they may not be explicitly defined or to evaluate processes that extend infinitely.
Ln(x): The natural logarithm, denoted as ln(x), is a logarithmic function that represents the power to which the base e must be raised to get the value x. The natural logarithm is a fundamental concept in calculus and is closely related to the topics of the Divergence and Integral Tests.
Monotonicity: Monotonicity is a property that describes the behavior of a function or sequence, indicating whether the values are consistently increasing, decreasing, or staying the same as the independent variable changes. This concept is particularly important in the study of sequences and the application of the Divergence and Integral Tests.
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.
Partial sum: A partial sum is the sum of the first $n$ terms in a sequence. It provides an approximation to the sum of an infinite series.
Partial Sum: A partial sum is the sum of the first n terms of an infinite series. It represents the accumulated value of the series up to a certain point, providing an approximation of the series' final sum as the number of terms increases.
Remainder estimate: A remainder estimate provides a bound on the error when approximating an infinite series by a partial sum. It helps determine how close the partial sum is to the actual value of the series.
Remainder Term: The remainder term, also known as the error term or the Lagrange remainder, is a mathematical concept that represents the difference between the actual value of a function and its approximation using a Taylor series or a Maclaurin series. It quantifies the error or the accuracy of the approximation, and is crucial in understanding the convergence and the applicability of these series expansions.
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.
Telescoping series: A telescoping series is an infinite series where most terms cancel out with subsequent terms, leaving only a few terms to sum. This characteristic makes it easier to find the series' sum.
Telescoping Series: A telescoping series is a type of infinite series where each term can be expressed as the difference between two consecutive terms in the series. This structure allows for the simplification and evaluation of the series through a process of successive cancellation of terms.
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5.4 Comparison Tests