Hi Mark!
Great question! The easiest way to approach this problem is with the Limit Comparison Test.
We want to determine whether the following series converges or diverges:
Σ (n² - 3n + 5)/(n³ + 4n - 1), from n = 1 to infinity.
Step 1: Look at the highest powers of n.
In the numerator, the highest power is n².
In the denominator, the highest power is n³.
For large values of n, the lower-degree terms become less important, so our expression behaves approximately like:
n²/n³ = 1/n
This is a helpful observation because we already know that the harmonic series, Σ(1/n), diverges.
Step 2: Apply the Limit Comparison Test.
Let:
aₙ = (n² - 3n + 5)/(n³ + 4n - 1)
bₙ = 1/n
We calculate:
L = lim(n→∞) (aₙ/bₙ)
Substituting:
L = lim(n→∞) [(n² - 3n + 5)/(n³ + 4n - 1)] / (1/n)
Simplify by multiplying by n:
L = lim(n→∞) (n³ - 3n² + 5n)/(n³ + 4n - 1)
Step 3: Evaluate the limit.
Divide the numerator and denominator by n³:
L = lim(n→∞) (1 - 3/n + 5/n²)/(1 + 4/n² - 1/n³)
As n approaches infinity, all terms containing 1/n, 1/n², or 1/n³ approach zero.
Therefore:
L = 1/1 = 1
Step 4: Interpret the result.
The Limit Comparison Test tells us that if L is a positive, finite number, both series have the same convergence behavior.
Since L = 1 and the harmonic series Σ(1/n) diverges, our original series must also diverge.
FINAL ANSWER: The series DIVERGES.
Here's a useful tip for future problems: When you have a rational expression involving powers of n, start by comparing the highest powers in the numerator and denominator. This often reveals which familiar series you should compare it to.
Hope this helps!
