(a) Show that the function f determined by the n th term of...

(a) Show that the function $f$ determined by the $n$ th term of the series satisfies the hypotheses of the integral test. (b) Use the integral test to determine whether the series converges or diverges.
$$
\sum_{n=1}^{\infty} \frac{1}{1+16 n^{2}}
$$

Key Concepts

Integral Test

The integral test for series convergence is a method that compares the sum of a series to the value of an improper integral. If a function f(x) corresponding to the series terms is continuous, positive, and monotonically decreasing for x greater than some value, then the convergence of the integral from that value to infinity implies convergence of the series, and divergence of the integral implies divergence of the series.

Improper Integral

An improper integral is an integral with limits of integration that are infinite or where the integrand has discontinuities. Determining whether an improper integral converges or diverges is crucial when applying the integral test, as the behavior of the integral directly indicates the behavior of the series.

Monotonic Decreasing Functions

For the integral test to be applicable, the function corresponding to the series must be monotonically decreasing. This property ensures that the function does not have fluctuations that could complicate the comparison between the series and the related integral, thereby simplifying the analysis of the convergence of both.

Series Convergence and Divergence

The concepts of convergence and divergence in the context of series refer to whether the sum of an infinite sequence of terms leads to a finite limit or not. Utilizing tests like the integral test allows one to determine the long-term behavior of the series by examining the corresponding function's integral.

Transcript

Please give Ace some feedback