Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.


g.The series ∑ (from k = 1 to ∞) (k² / (k² + 1)) converges.

Verified step by step guidance

1. Identify the general term of the series: \( a_k = \frac{k^2}{k^2 + 1} \).
2. Examine the behavior of \( a_k \) as \( k \to \infty \). Calculate the limit \( \lim_{k \to \infty} a_k = \lim_{k \to \infty} \frac{k^2}{k^2 + 1} \).
3. Since the degrees of numerator and denominator are the same, the limit is the ratio of the leading coefficients, which is \( 1 \). So, \( \lim_{k \to \infty} a_k = 1 \).
4. Recall the Divergence Test (also called the Test for Divergence): if \( \lim_{k \to \infty} a_k \neq 0 \), then the series \( \sum a_k \) diverges.
5. Because \( \lim_{k \to \infty} a_k = 1 \neq 0 \), the series \( \sum_{k=1}^\infty \frac{k^2}{k^2 + 1} \) does not converge; it diverges.

Key Concepts

Definition of Series Convergence

A series converges if the sequence of its partial sums approaches a finite limit as the number of terms goes to infinity. If the partial sums do not approach a finite value, the series diverges. Understanding this helps determine whether the given infinite sum converges or not.

Term Test for Divergence

If the terms of a series do not approach zero as k approaches infinity, the series must diverge. This is a quick test to check convergence; if the limit of the general term is not zero, the series cannot converge.

Behavior of the General Term (k² / (k² + 1))

Analyzing the limit of the term k² / (k² + 1) as k approaches infinity shows it approaches 1, not zero. Since the terms do not tend to zero, this indicates the series ∑ (k² / (k² + 1)) diverges by the term test.