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Explain the conditions required to apply the Integral Test.
The function $f(x)$ must be continuous, positive, and decreasing on the interval $[k, \infty)$, and $a_n = f(n)$.
Explain why the function must be positive for the Integral Test.
If the function is not positive, the comparison between the integral and the series is not valid, as the areas and sums could have different signs.
Explain why the function must be decreasing for the Integral Test.
If the function is not decreasing, the integral may not accurately represent the sum of the series, as the terms may not consistently get smaller.
Explain the relationship between the convergence of an integral and the convergence of a series in the Integral Test.
If the improper integral converges, the corresponding infinite series also converges. If the improper integral diverges, the corresponding infinite series also diverges.
What does it mean for an improper integral to converge?
The limit of the integral as the upper bound approaches infinity exists and is a finite number.
What does it mean for an improper integral to diverge?
The limit of the integral as the upper bound approaches infinity does not exist or is infinite.
Explain the role of u-substitution in the Integral Test.
U-substitution is a technique used to simplify the integral, making it easier to evaluate and determine its convergence or divergence.
Explain why the starting index 'k' matters in the Integral Test.
The starting index 'k' determines the lower limit of integration and the first term of the series. The conditions of the Integral Test must hold for $x \geq k$.
Explain what happens if the conditions for the integral test are not met.
The integral test cannot be applied, and another test for convergence or divergence must be used.
Explain the purpose of the integral test.
To determine whether an infinite series converges or diverges by comparing it to an improper integral.
If the graph of $f(x)$ is always above the x-axis and decreasing, what does this imply for the Integral Test?
It suggests that $f(x)$ is positive and decreasing, which are two conditions needed to apply the Integral Test.
How can you visually determine if a function is decreasing from its graph?
The graph should be going downwards as you move from left to right.
If the area under the curve of $f(x)$ from $k$ to $\infty$ is finite, what does this mean for the series $\sum a_n$?
It suggests that the series $\sum a_n$ converges, according to the Integral Test.
How does the graph of the derivative of $f(x)$ help in the Integral Test?
If $f'(x) < 0$ for $x \geq k$, then $f(x)$ is decreasing, a condition for the Integral Test.
How can the graph of $f(x) = \frac{1}{x}$ be used to illustrate the Integral Test?
The graph shows a positive, decreasing function. The area under the curve from 1 to infinity can be compared to the sum of the series $\sum \frac{1}{n}$.
What does the area under the curve of $f(x)$ represent in the context of the Integral Test?
It represents the value of the improper integral $\int_k^{\infty} f(x) \, dx$, which is compared to the sum of the series.
How does the steepness of the graph of $f(x)$ relate to the convergence of the series?
A steeper decreasing graph suggests faster convergence, while a shallower decreasing graph may lead to divergence.
What does a graph that oscillates above and below the x-axis imply for the Integral Test?
The Integral Test cannot be applied since the function must be positive.
How does a discontinuity in the graph of $f(x)$ affect the Integral Test?
If the discontinuity occurs within the interval of integration, the Integral Test cannot be directly applied.
How does the graph of $f(x) = \frac{1}{x^2}$ illustrate the Integral Test?
The graph is positive and decreasing. The area under the curve from 1 to infinity converges, illustrating the convergence of the series $\sum \frac{1}{n^2}$.
What is the general form of the Integral Test?
If $f(x)$ is positive, continuous, and decreasing on $[k, \infty)$ and $a_n = f(n)$, then $\sum_{n=k}^{\infty} a_n$ converges if and only if $\int_k^{\infty} f(x) \, dx$ converges.
What is the integral test formula for convergence?
If $\int_k^{\infty} f(x) \, dx$ converges, then $\sum_{n=k}^{\infty} a_n$ converges.
What is the integral test formula for divergence?
If $\int_k^{\infty} f(x) \, dx$ diverges, then $\sum_{n=k}^{\infty} a_n$ diverges.
What is the integral of $\frac{1}{x}$?
$\int \frac{1}{x} \, dx = \ln|x| + C$
What is the integral of $\frac{1}{1+x^2}$?
$\int \frac{1}{1+x^2} \, dx = \arctan(x) + C$
What is the formula for u-substitution?
$\int f(g(x))g'(x) \, dx = \int f(u) \, du$, where $u = g(x)$ and $du = g'(x) \, dx$
What is the formula for the integral of a constant?
$\int c \, dx = cx + C$
Give the formula for integration by substitution.
$\int f'(g(x))g'(x)dx = f(g(x)) + C$
What is the formula for the arctangent integral?
$\int \frac{1}{a^2 + x^2} dx = \frac{1}{a}arctan(\frac{x}{a}) + C$
What is the formula for the natural logarithm integral?
$\int \frac{1}{x} dx = ln|x| + C$