Where a power series converges, found with the ratio test. 70 problems with worked solutions; in 70 of them every
equation is proved by a computer algebra system.
15 worked problems
Try each one before opening the solution. Every line is labeled with what checked it. How we verify
The series converges when 1/3|x − 1| < 1, that is |x − 1| < 3.
Reviewed
Answer \( R = 3 \)
✓ 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
The series converges when 1|x − 2| < 1, that is |x − 2| < 1.
Reviewed
Answer \( R = 1 \)
✓ 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
The series converges when 1/5|x − -3| < 1, that is |x − -3| < 5.
Reviewed
Answer \( R = 5 \)
✓ 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
The series converges when 1/3|x − 2| < 1, that is |x − 2| < 3.
Reviewed
Answer \( R = 3 \)
✓ 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
The series converges when 1|x − 3| < 1, that is |x − 3| < 1.
Reviewed
Answer \( R = 1 \)
✓ 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
The series converges when 1/5|x − 3| < 1, that is |x − 3| < 5.
Reviewed
Answer \( R = 5 \)
✓ 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
The series converges when 1/4|x − 0| < 1, that is |x − 0| < 4.
Reviewed
Answer \( R = 4 \)
✓ 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
The series converges when 1/5|x − -3| < 1, that is |x − -3| < 5.
Reviewed
Answer \( R = 5 \)
✓ 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
Find the radius of convergence of \( \displaystyle \sum_{n=1}^{\infty} \frac{4^{- n} x^{n}}{n} \).
Use the ratio test on the absolute values of the terms.
Reviewed
\[ \lim_{n \to \infty}\left(\frac{4^{n} 4^{- n - 1} n x^{- n} x^{n + 1}}{x \left(n + 1\right)}\right) = \frac{1}{4} \]
|a_(n+1)/a_n| → 1/4·|x − 0|.✓ Proved
The series converges when 1/4|x − 0| < 1, that is |x − 0| < 4.
Reviewed
Answer \( R = 4 \)
✓ 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
The series converges when 1/5|x − 0| < 1, that is |x − 0| < 5.
Reviewed
Answer \( R = 5 \)
✓ 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
The series converges when 1|x − 2| < 1, that is |x − 2| < 1.
Reviewed
Answer \( R = 1 \)
✓ 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
The series converges when 1/4|x − -2| < 1, that is |x − -2| < 4.
Reviewed
Answer \( R = 4 \)
✓ 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
The series converges when 1|x − -2| < 1, that is |x − -2| < 1.
Reviewed
Answer \( R = 1 \)
✓ 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
The series converges when 1|x − -1| < 1, that is |x − -1| < 1.
Reviewed
Answer \( R = 1 \)
✓ 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
The series converges when 1|x − 0| < 1, that is |x − 0| < 1.
Reviewed
Answer \( R = 1 \)
✓ 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