91Ó°ÊÓ

To determine whether a series converges, one common and effective method is the Comparison Test. This test helps us analyze whether a series converges by comparing it with another series that has known convergence properties.
Let's break it down:

• Suppose we have two series, \( ext{Series A} = \sum_{k=1}^{\infty} a_k \) and \( ext{Series B} = \sum_{k=1}^{\infty} b_k \).
• If \( 0 \leq a_k \leq b_k \) for all values of \( k \) beyond some index, and if Series B is known to converge, then Series A also converges.
• On the flip side, if \( a_k \geq b_k \geq 0 \) and Series B diverges, so does Series A.

In the exercise, recognizing that \( an^{-1} k \) is bounded helps us establish an upper boundary for the terms. Comparing \( \frac{\tan^{-1} k}{1+k^2} \) with the series \( \sum_{k=1}^{\infty}\frac{1}{k^2} \) allows us to conclude convergence since \( \sum_{k=1}^{\infty}\frac{1}{k^2} \) is a convergent p-series.

In mathematics, understanding bounded functions can simplify complex series analysis. A function like \( \tan^{-1} k \) is an excellent example, as its values lie within a fixed range.
Why is this important? Imagine trying to understand the behavior of a series that seems complex at first glance. Knowing that certain elements are bounded can dramatically simplify your work.
A bounded function has values that never exceed a particular upper limit and never go below a specific lower limit. For instance:

• For \( \tan^{-1} k \), as \( k \) approaches infinity, the value tends toward \( \frac{\pi}{2} \), making it bounded above by \( \frac{\pi}{2} \).
• This predictable behavior implies that for the given series, \( \frac{\tan^{-1} k}{1+k^2} \) can never exceed the bound \( \frac{\pi}{2(1+k^2)} \).

By recognizing this, we can smartly compare series terms to make judgments on convergence, as in using the Comparison Test.

The P-Series is a critical concept in understanding series convergence, specifically related to the behavior of the series \( \sum_{k=1}^{\infty} \frac{1}{k^p} \). With this series type:

• Convergence depends critically on the value of \( p \).
• It converges when \( p > 1 \) and diverges when \( p \leq 1 \).

Given this powerful test, examining a series like \( \sum_{k=1}^{\infty} \frac{1}{k^2} \) confirms its convergence, as \( p = 2 > 1 \). This type of series serves as a baseline in many exercises, allowing us to compare and determine the convergence from known outcomes.
In this exercise, the \( \sum_{k=1}^{\infty} \frac{1}{k^2} \) series functions as a classic example of how P-series provide straightforward benchmarks. Recognizing a series as a P-series can immediately inform about its convergence, letting you apply techniques like the Comparison Test efficiently.