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 $du = g'(x) , dx$

$\int c , dx = cx + C$

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

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

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

It suggests that $f(x)$ is positive and decreasing, which are two conditions needed to apply the Integral Test.

The graph should be going downwards as you move from left to right.

It suggests that the series $\sum a_n$ converges, according to the Integral Test.

If $f'(x) < 0$ for $x \geq k$, then $f(x)$ is decreasing, a condition for 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 $\sum \frac{1}{n}$.

It represents the value of the improper integral $\int_k^{\infty} f(x) , dx$, which is compared to the sum of the series.

A steeper decreasing graph suggests faster convergence, while a shallower decreasing graph may lead to divergence.

The Integral Test cannot be applied since the function must be positive.

If the discontinuity occurs within the interval of integration, the Integral Test cannot be directly applied.

The graph is positive and decreasing. The area under the curve from 1 to infinity converges, illustrating the convergence of the series $\sum \frac{1}{n^2}$.

1. Verify $f(x) = \frac{1}{x^2+1}$ is positive, continuous, and decreasing for $x \geq 1$. 2. Evaluate $\int_1^{\infty} \frac{1}{x^2+1} , dx$. 3. If the integral converges, the series converges; if it diverges, the series diverges.

1. Use u-substitution: $u = \ln(x)$, $du = \frac{1}{x} , dx$. 2. Rewrite the integral as $\int \frac{1}{u} , du$. 3. Evaluate to get $\ln|u| + C = \ln|\ln(x)| + C$. 4. Evaluate the improper integral.

Show that the derivative $f'(x) = -\frac{1}{x^2}$ is negative for $x > 0$.

1. Verify $f(x)$ is positive, continuous, and decreasing for $x \geq k$. 2. Evaluate the improper integral $\int_k^{\infty} f(x) , dx$. 3. Determine if the integral converges or diverges. 4. Conclude the series converges/diverges accordingly.

Choose $u = \ln(n)$ because its derivative $\frac{1}{n}$ is also present in the series/integral.

Compute the derivative of the function. If the derivative is negative over the interval of interest, the function is decreasing.

Replace the infinite limit with a variable, say $b$, and take the limit as $b$ approaches infinity.

Apply the integral test with $f(x) = \frac{1}{x^p}$. The series converges if $p > 1$ and diverges if $p \leq 1$.

Look for a function $f(x)$ such that $f(n)$ matches the terms of the series and $f(x)$ is positive, continuous, and decreasing.

Ensure that the function $f(x)$ satisfies the conditions of the Integral Test for all $x$ greater than or equal to the lower limit.