Divergence, p-series, integral, comparison and ratio tests: which one, and why. 66 problems with worked solutions; in 66 of them every
equation is proved by a computer algebra system.
14 worked problems
Try each one before opening the solution. Every line is labeled with what checked it. How we verify
The limit is 0 < 1, so the series converges.✓ Proved
Answer \( \text{converges} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Bars show the terms n²/3^n for n = 1 to 24; dots show the running totals. The totals level off: the series converges.
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Σ 1/n^2 converges (p = 2), so the given series does too.
Reviewed
Answer \( \text{converges} \)
Bars show the terms (n + 1)/(n³ + n + 1) for n = 1 to 24; dots show the running totals. The totals level off: the series converges.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Does \( \displaystyle \sum_{n=1}^{\infty} \frac{4 n + 1}{n^{3} + n + 1} \) converge or diverge?
For large n the terms behave like 4/n^2: compare with the p-series Σ 1/n^2.
Reviewed
\[ \lim_{n \to \infty}\left(\frac{n^{2} \left(4 n + 1\right)}{n^{3} + n + 1}\right) = 4 \]
The limit is 4, positive and finite.✓ Proved
Σ 1/n^2 converges (p = 2), so the given series does too.
Reviewed
Answer \( \text{converges} \)
Bars show the terms (4n + 1)/(n³ + n + 1) for n = 1 to 24; dots show the running totals. The totals level off: the series converges.
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Bars show the terms n/5^n for n = 1 to 24; dots show the running totals. The totals level off: the series converges.
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Does \( \displaystyle \sum_{n=1}^{\infty} \frac{1}{n^{\frac{3}{2}}} \) converge or diverge?
This is a p-series with p = 3/2.
Reviewed
A p-series converges exactly when p > 1, so it converges.
Reviewed
Answer \( \text{converges} \)
Bars show the terms n^(-3/2) for n = 1 to 24; dots show the running totals. The totals level off: the series converges.
✓ Nihil obstat Lines: 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Does \( \displaystyle \sum_{n=1}^{\infty} \frac{4 n + 1}{5 n + 3} \) converge or diverge?
\[ \lim_{n \to \infty}\left(\frac{4 n + 1}{5 n + 3}\right) = \frac{4}{5} \]
The terms do not go to 0...✓ Proved
...so by the divergence test the series diverges.
Reviewed
Answer \( \text{diverges} \)
Bars show the terms (4n + 1)/(5n + 3) for n = 1 to 24; dots show the running totals. The totals never settle: the series diverges.
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Bars show the terms (n + 1)/(5n + 3) for n = 1 to 24; dots show the running totals. The totals never settle: the series diverges.
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Does \( \displaystyle \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \) converge or diverge?
This is a p-series with p = 1/2.
Reviewed
A p-series converges exactly when p > 1, so it diverges.
Reviewed
Answer \( \text{diverges} \)
Bars show the terms 1/sqrt(n) for n = 1 to 24; dots show the running totals. The totals never settle: the series diverges.
✓ Nihil obstat Lines: 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Bars show the terms (3n + 1)/(n + 3) for n = 1 to 24; dots show the running totals. The totals never settle: the series diverges.
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Does \( \displaystyle \sum_{n=1}^{\infty} \frac{1}{n} \) converge or diverge?
This is a p-series with p = 1.
Reviewed
A p-series converges exactly when p > 1, so it diverges.
Reviewed
Answer \( \text{diverges} \)
Bars show the terms 1/n for n = 1 to 24; dots show the running totals. The totals never settle: the series diverges.
✓ Nihil obstat Lines: 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Bars show the terms (n + 1)/(n + 3) for n = 1 to 24; dots show the running totals. The totals never settle: the series diverges.
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify