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.

If $\int_k^{\infty} f(x) , dx$ converges, then $\sum_{n=k}^{\infty} a_n$ converges.

If $\int_k^{\infty} f(x) , dx$ diverges, then $\sum_{n=k}^{\infty} a_n$ diverges.

$\int \frac{1}{x} , dx = \ln|x| + C$

$\int \frac{1}{1+x^2} , dx = \arctan(x) + C$

$\int f(g(x))g'(x) , dx = \int f(u) , du$, where $u = g(x)$ and $du = g'(x) , dx$

$\int c , dx = cx + C$

$\int f'(g(x))g'(x)dx = f(g(x)) + C$

$\int \frac{1}{a^2 + x^2} dx = \frac{1}{a}arctan(\frac{x}{a}) + C$

$\int \frac{1}{x} dx = ln|x| + C$

An improper integral is convergent if it evaluates to a finite number.

An improper integral is divergent if it evaluates to $\pm\infty$ or does not exist.

A function $f(x)$ is positive and decreasing over an interval if $f(x) > 0$ and $f'(x) < 0$ for all $x$ in that interval.

$a_n = f(n)$ means the terms of the series are obtained by evaluating the function $f(x)$ at integer values $n$.

A method to determine the convergence or divergence of an infinite series by comparing it to an improper integral.

The sum of an infinite number of terms.

An integral with infinite limits of integration or a discontinuous integrand.

The sum of the infinite series approaches a finite value.

The sum of the infinite series does not approach a finite value.

A technique for evaluating integrals by substituting a function $u$ for part of the integrand.

The function $f(x)$ must be continuous, positive, and decreasing on the interval $[k, \infty)$, and $a_n = f(n)$.

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.

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.

If the improper integral converges, the corresponding infinite series also converges. If the improper integral diverges, the corresponding infinite series also diverges.

The limit of the integral as the upper bound approaches infinity exists and is a finite number.

The limit of the integral as the upper bound approaches infinity does not exist or is infinite.

U-substitution is a technique used to simplify the integral, making it easier to evaluate and determine its convergence or divergence.

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 $x \geq k$.

The integral test cannot be applied, and another test for convergence or divergence must be used.

To determine whether an infinite series converges or diverges by comparing it to an improper integral.