By Fatskills Exam Guides Team — the exam nerds behind 28,500+ quizzes and 2.1M practice questions across 500+ global exams.
The Integral, Comparison, and Ratio Tests are methods used to determine whether a series of numbers is convergent (its sum approaches a finite value) or divergent (its sum grows without bound). These tests are essential in understanding the behavior of infinite series, which are crucial in mathematics, physics, engineering, and economics.
Infinite series appear in many real-world contexts, such as: * Modeling population growth or decay in biology and economics * Calculating the sum of an infinite geometric series to find the present value of an annuity in finance * Approximating functions using Taylor series in calculus and physics * Analyzing the stability of systems in control theory and signal processing
A series $\sum_{n=1}^{\infty} a_n$ is said to be convergent if the sequence of partial sums $S_n = \sum_{k=1}^{n} a_k$ converges to a finite limit $S$ as $n \to \infty$. Otherwise, the series is divergent.
The Integral Test states that if $f(x)$ is a continuous, positive, and decreasing function on $[1, \infty)$, then the series $\sum_{n=1}^{\infty} f(n)$ and the improper integral $\int_{1}^{\infty} f(x) dx$ either both converge or both diverge.
The Comparison Test states that if ${a_n}$ and ${b_n}$ are sequences of positive numbers, and $a_n \leq b_n$ for all $n$, then: * If $\sum b_n$ converges, then $\sum a_n$ converges. * If $\sum a_n$ diverges, then $\sum b_n$ diverges.
The Ratio Test states that if ${a_n}$ is a sequence of positive numbers, then: * If $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1$, then $\sum a_n$ converges. * If $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| > 1$, then $\sum a_n$ diverges. * If $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 1$, the test is inconclusive.
To determine whether a series converges or diverges using the Integral, Comparison, and Ratio Tests:
Evaluate the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ using the Integral Test.
$$\int_{1}^{\infty} \frac{1}{x^2} dx = \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^2} dx = \lim_{b \to \infty} \left( -\frac{1}{x} \right) \Big|{1}^{b} = \lim + 1 \right) = 1$$} \left( -\frac{1}{b
The series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges.
Determine whether the series $\sum_{n=1}^{\infty} \frac{1}{n}$ converges or diverges by comparing it with the harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$.
Since the harmonic series diverges, the series $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges.
Evaluate the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{2^n}$ using the Ratio Test.
$$\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{1}{2^{n+1}} \cdot \frac{2^n}{1} \right| = \lim_{n \to \infty} \frac{1}{2} = \frac{1}{2} < 1$$
The series $\sum_{n=1}^{\infty} \frac{1}{2^n}$ converges.
Which of the following series converges using the Integral Test?
A) $\sum_{n=1}^{\infty} \frac{1}{n}$ B) $\sum_{n=1}^{\infty} \frac{1}{n^2}$ C) $\sum_{n=1}^{\infty} \frac{1}{n^3}$ D) $\sum_{n=1}^{\infty} \frac{1}{n^4}$
B) $\sum_{n=1}^{\infty} \frac{1}{n^2}$
The series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges using the Integral Test, since the improper integral $\int_{1}^{\infty} \frac{1}{x^2} dx$ converges.
A) The harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges, so it is not a good candidate for the Integral Test. C) The series $\sum_{n=1}^{\infty} \frac{1}{n^3}$ converges, but it is not the best example for the Integral Test. D) The series $\sum_{n=1}^{\infty} \frac{1}{n^4}$ converges, but it is not the best example for the Integral Test.
Which of the following series diverges using the Comparison Test?
A) $\sum_{n=1}^{\infty} \frac{1}{n^2}$ B) $\sum_{n=1}^{\infty} \frac{1}{n}$ C) $\sum_{n=1}^{\infty} \frac{1}{n^3}$ D) $\sum_{n=1}^{\infty} \frac{1}{n^4}$
B) $\sum_{n=1}^{\infty} \frac{1}{n}$
The series $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges, so it is a good candidate for the Comparison Test. Since the harmonic series diverges, the series $\sum_{n=1}^{\infty} \frac{1}{n}$ also diverges.
A) The series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges, so it is not a good candidate for the Comparison Test. C) The series $\sum_{n=1}^{\infty} \frac{1}{n^3}$ converges, so it is not a good candidate for the Comparison Test. D) The series $\sum_{n=1}^{\infty} \frac{1}{n^4}$ converges, so it is not a good candidate for the Comparison Test.
Which of the following series converges using the Ratio Test?
A) $\sum_{n=1}^{\infty} \frac{1}{n}$ B) $\sum_{n=1}^{\infty} \frac{1}{n^2}$ C) $\sum_{n=1}^{\infty} \frac{1}{2^n}$ D) $\sum_{n=1}^{\infty} \frac{1}{n^3}$
C) $\sum_{n=1}^{\infty} \frac{1}{2^n}$
The series $\sum_{n=1}^{\infty} \frac{1}{2^n}$ converges using the Ratio Test, since the limit $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \frac{1}{2} < 1$.
A) The harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges, so it is not a good candidate for the Ratio Test. B) The series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges, but it is not a good example for the Ratio Test. D) The series $\sum_{n=1}^{\infty} \frac{1}{n^3}$ converges, but it is not a good example for the Ratio Test.
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