Test the series for convergence or divergence: $\displaystyle\sum_{n=1}^\infty \frac{n^5 - 1}{n^6 + 1}.$

Neetesh Kumar | December 23, 2024

Calculus Homework Help

This is the solution to Math 1c
Assignment: 11.7 Question Number 1
Contact me if you need help with Homework, Assignments, Tutoring Sessions, or Exams for STEM subjects.
You can see our Testimonials or Vouches from here of the previous works I have done.

Get Homework Help

---

Step-by-step solution:

Step 1: Analyze the general term of the series:

The general term of the series is:

$a_n = \frac{n^5 - 1}{n^6 + 1}.$

As $n$ becomes large, the highest powers of $n$ dominate both the numerator and denominator.

To determine the behavior of the series, simplify $a_n$ for large $n$:

$a_n \sim \frac{n^5}{n^6} = \frac{1}{n}.$

This suggests that $a_n$ behaves like the harmonic series term $\frac{1}{n}$, which is divergent.

Step 2: Apply the Limit Comparison Test:

We compare $a_n$ with the harmonic series $\frac{1}{n}$, which diverges.

The Limit Comparison Test states that if:

$\displaystyle\lim_{n \to \infty} \frac{a_n}{b_n} = L,$

where $0 < L < \infty$, then both series either converge or diverge together.

Let $b_n = \frac{1}{n}$. Compute the limit:

$\displaystyle\lim_{n \to \infty} \frac{\frac{n^5 - 1}{n^6 + 1}}{\frac{1}{n}} = \displaystyle\lim_{n \to \infty} \frac{n^5 - 1}{n^6 + 1} \cdot n.$

Simplify:

$\displaystyle\lim_{n \to \infty} \frac{n^{6} - n}{n^6 + 1}.$

Divide numerator and denominator by $n^6$:

$\displaystyle\lim_{n \to \infty} \frac{1 - \frac{1}{n^5}}{1 + \frac{1}{n^6}} = \frac{1 - 0}{1 + 0} = 1.$

Since the limit is $L = 1$, and $L$ is finite and positive, the Limit Comparison Test applies.

Step 3: Conclusion:

Since the harmonic series $\sum \frac{1}{n}$ diverges, and the given series behaves similarly for large $n$, the series:

$\displaystyle\sum_{n=1}^\infty \frac{n^5 - 1}{n^6 + 1}$

is divergent.

Final Answer:

The series is: $\boxed{\text{divergent}}$

Please comment below if you find any error in this solution.
If this solution helps, then please share this with your friends.
Please subscribe to my Youtube channel for video solutions to similar questions.
Keep Smiling :-)