Math Problem Statement

Determine convergence or divergence of the series ∑ (5n^2 + 4) / (n^2 + 7n + 1) from n=1 to ∞.

Solution

The given problem asks to determine whether the series

n=15n2+4n2+7n+1\sum_{n=1}^{\infty} \frac{5n^2 + 4}{n^2 + 7n + 1}

converges or diverges.


Solution:

Step 1: Analyze the general term

The general term of the series is:

an=5n2+4n2+7n+1.a_n = \frac{5n^2 + 4}{n^2 + 7n + 1}.

For large values of nn, the highest power of nn in the numerator and denominator dominates. Simplifying the leading terms:

an5n2n2=5.a_n \sim \frac{5n^2}{n^2} = 5.

Step 2: Test for divergence

If limnan0\lim_{n \to \infty} a_n \neq 0, the series diverges by the Divergence Test (also called the nn-th term test for divergence).

Compute the limit:

limnan=limn5n2+4n2+7n+1.\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{5n^2 + 4}{n^2 + 7n + 1}.

Divide numerator and denominator by n2n^2:

limnan=5+4n21+7n+1n2.\lim_{n \to \infty} a_n = \frac{5 + \frac{4}{n^2}}{1 + \frac{7}{n} + \frac{1}{n^2}}.

As nn \to \infty, the terms 4n2\frac{4}{n^2}, 7n\frac{7}{n}, and 1n2\frac{1}{n^2} approach 00. Thus:

limnan=51=5.\lim_{n \to \infty} a_n = \frac{5}{1} = 5.

Since limnan=50\lim_{n \to \infty} a_n = 5 \neq 0, the series diverges.


Conclusion:

The series diverges.


Would you like further explanation or related examples? Here are five follow-up questions to explore:

  1. What is the Divergence Test, and why is it applicable here?
  2. How do we determine convergence for series where the Divergence Test is inconclusive?
  3. Can the Limit Comparison Test be applied to this series for additional verification?
  4. How does the presence of dominant terms in the numerator and denominator affect convergence?
  5. What is an example of a series that converges, and how does it differ from this one?

Tip: For rational functions in series, always check the leading terms as nn \to \infty to simplify your analysis!

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Math Problem Analysis

Mathematical Concepts

Series Convergence
Divergence Test
Asymptotic Analysis

Formulas

Limit of a ratio of polynomials
Divergence Test

Theorems

Divergence Test

Suitable Grade Level

Grades 11-12 or college-level calculus