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The root test practice problems

Decide convergence with the root test: the limit of the n-th root of |aₙ|. 60 problems with worked solutions; in 60 of them every equation is proved by a computer algebra system.

9 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 7.569

medium
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{4 n + 3}{5 n + 2}\right)^{n} \) converges.
  1. The root test looks at lim |aₙ|^(1/n): below 1 converges, above 1 diverges.
    Reviewed
  2. \[ \left(\left(\frac{4 n + 3}{5 n + 2}\right)^{n}\right)^{1 \frac{1}{n}} = \frac{4 n + 3}{5 n + 2} \]
    The n-th root.✓ Proved
  3. \[ \lim_{n \to \infty}\left(\frac{4 n + 3}{5 n + 2}\right) = \frac{4}{5} \]
    The limit is 4/5, < 1.✓ Proved
  4. So the series converges.
    Reviewed
Answer \( \lim \sqrt[n]{|a_n|} = \frac{4}{5},\ \text{converges} \)

✓ Nihil obstat Lines: 2 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

Problem 7.570

medium
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{n}{n + 3}\right)^{n^{2}} \) converges.
  1. The root test looks at lim |aₙ|^(1/n): below 1 converges, above 1 diverges.
    Reviewed
  2. \[ \left(\left(\frac{n}{n + 3}\right)^{n^{2}}\right)^{1 \frac{1}{n}} = \left(\frac{n}{n + 3}\right)^{n} \]
    The n-th root.✓ Proved
  3. \[ \lim_{n \to \infty} \left(\frac{n}{n + 3}\right)^{n} = e^{-3} \]
    The limit is exp(-3), < 1.✓ Proved
  4. So the series converges.
    Reviewed
Answer \( \lim \sqrt[n]{|a_n|} = e^{-3},\ \text{converges} \)

✓ Nihil obstat Lines: 2 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

Problem 7.571

medium
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{2 n}{4 n + 1}\right)^{n} \) converges.
  1. The root test looks at lim |aₙ|^(1/n): below 1 converges, above 1 diverges.
    Reviewed
  2. \[ \left(\left(\frac{2 n}{4 n + 1}\right)^{n}\right)^{1 \frac{1}{n}} = \frac{2 n}{4 n + 1} \]
    The n-th root.✓ Proved
  3. \[ \lim_{n \to \infty}\left(\frac{2 n}{4 n + 1}\right) = \frac{1}{2} \]
    The limit is 1/2, < 1.✓ Proved
  4. So the series converges.
    Reviewed
Answer \( \lim \sqrt[n]{|a_n|} = \frac{1}{2},\ \text{converges} \)

✓ Nihil obstat Lines: 2 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

Problem 7.573

medium
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} 5^{- n} n \) converges.
  1. The root test looks at lim |aₙ|^(1/n): below 1 converges, above 1 diverges.
    Reviewed
  2. \[ \left(5^{- n} n\right)^{\frac{1}{n}} = \frac{n^{\frac{1}{n}}}{5} \]
    The n-th root.✓ Proved
  3. \[ \lim_{n \to \infty}\left(\frac{n^{\frac{1}{n}}}{5}\right) = \frac{1}{5} \]
    The limit is 1/5, < 1.✓ Proved
  4. So the series converges.
    Reviewed
Answer \( \lim \sqrt[n]{|a_n|} = \frac{1}{5},\ \text{converges} \)

✓ Nihil obstat Lines: 2 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

Problem 7.575

medium
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} 2^{- n} n^{3} \) converges.
  1. The root test looks at lim |aₙ|^(1/n): below 1 converges, above 1 diverges.
    Reviewed
  2. \[ \left(2^{- n} n^{3}\right)^{\frac{1}{n}} = \frac{n^{\frac{3}{n}}}{2} \]
    The n-th root.✓ Proved
  3. \[ \lim_{n \to \infty}\left(\frac{n^{\frac{3}{n}}}{2}\right) = \frac{1}{2} \]
    The limit is 1/2, < 1.✓ Proved
  4. So the series converges.
    Reviewed
Answer \( \lim \sqrt[n]{|a_n|} = \frac{1}{2},\ \text{converges} \)

✓ Nihil obstat Lines: 2 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

Problem 7.578

medium
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(1 + \frac{1}{n}\right)^{n^{2}} \) converges.
  1. The root test looks at lim |aₙ|^(1/n): below 1 converges, above 1 diverges.
    Reviewed
  2. \[ \left(\left(1 + 1 \frac{1}{n}\right)^{n^{2}}\right)^{1 \frac{1}{n}} = \left(1 + \frac{1}{n}\right)^{n} \]
    The n-th root.✓ Proved
  3. \[ \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n} = e \]
    The limit is E, > 1.✓ Proved
  4. So the series diverges.
    Reviewed
Answer \( \lim \sqrt[n]{|a_n|} = e,\ \text{diverges} \)

✓ Nihil obstat Lines: 2 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

Problem 7.579

medium
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{n + 1}{3 n + 1}\right)^{n} \) converges.
  1. The root test looks at lim |aₙ|^(1/n): below 1 converges, above 1 diverges.
    Reviewed
  2. \[ \left(\left(\frac{n + 1}{3 n + 1}\right)^{n}\right)^{1 \frac{1}{n}} = \frac{n + 1}{3 n + 1} \]
    The n-th root.✓ Proved
  3. \[ \lim_{n \to \infty}\left(\frac{n + 1}{3 n + 1}\right) = \frac{1}{3} \]
    The limit is 1/3, < 1.✓ Proved
  4. So the series converges.
    Reviewed
Answer \( \lim \sqrt[n]{|a_n|} = \frac{1}{3},\ \text{converges} \)

✓ Nihil obstat Lines: 2 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

Problem 7.697

medium
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{3 n + 2}{n + 4}\right)^{n} \) converges.
  1. The root test looks at lim |aₙ|^(1/n): below 1 converges, above 1 diverges.
    Reviewed
  2. \[ \left(\left(\frac{3 n + 2}{n + 4}\right)^{n}\right)^{1 \frac{1}{n}} = \frac{3 n + 2}{n + 4} \]
    The n-th root.✓ Proved
  3. \[ \lim_{n \to \infty}\left(\frac{3 n + 2}{n + 4}\right) = 3 \]
    The limit is 3, > 1.✓ Proved
  4. So the series diverges.
    Reviewed
Answer \( \lim \sqrt[n]{|a_n|} = 3,\ \text{diverges} \)

✓ Nihil obstat Lines: 2 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

Problem 7.721

medium
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{n}{2 n + 3}\right)^{n} \) converges.
  1. The root test looks at lim |aₙ|^(1/n): below 1 converges, above 1 diverges.
    Reviewed
  2. \[ \left(\left(\frac{n}{2 n + 3}\right)^{n}\right)^{1 \frac{1}{n}} = \frac{n}{2 n + 3} \]
    The n-th root.✓ Proved
  3. \[ \lim_{n \to \infty}\left(\frac{n}{2 n + 3}\right) = \frac{1}{2} \]
    The limit is 1/2, < 1.✓ Proved
  4. So the series converges.
    Reviewed
Answer \( \lim \sqrt[n]{|a_n|} = \frac{1}{2},\ \text{converges} \)

✓ Nihil obstat Lines: 2 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

More the root test practice

Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{5 n}{2 n + 3}\right)^{n} \) converges.
Problem 7.572medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{5 n}{2 n + 1}\right)^{n} \) converges.
Problem 7.574medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} 5^{- n} n^{4} \) converges.
Problem 7.576medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{3 n + 2}{5 n + 4}\right)^{n} \) converges.
Problem 7.577medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{4 n + 1}{5 n + 4}\right)^{n} \) converges.
Problem 7.580medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(1 + \frac{3}{n}\right)^{n^{2}} \) converges.
Problem 7.581medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{2 n + 2}{5 n + 4}\right)^{n} \) converges.
Problem 7.582medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{5 n + 1}{3 n + 3}\right)^{n} \) converges.
Problem 7.583medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} 3^{- n} n \) converges.
Problem 7.639medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} 3^{- n} n^{2} \) converges.
Problem 7.640medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{n}{n + 1}\right)^{n^{2}} \) converges.
Problem 7.641medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(1 + \frac{2}{n}\right)^{n^{2}} \) converges.
Problem 7.642medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{2 n + 3}{4 n + 3}\right)^{n} \) converges.
Problem 7.643medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} 3^{- n} n^{4} \) converges.
Problem 7.644medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{4 n + 1}{3 n + 3}\right)^{n} \) converges.
Problem 7.645medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{5 n + 2}{4 n + 2}\right)^{n} \) converges.
Problem 7.646medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{n + 3}{3 n + 4}\right)^{n} \) converges.
Problem 7.647medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{4 n + 2}{5 n + 3}\right)^{n} \) converges.
Problem 7.648medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} 5^{- n} n^{2} \) converges.
Problem 7.649medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{5 n + 2}{3 n + 4}\right)^{n} \) converges.
Problem 7.650medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{4 n}{3 n + 3}\right)^{n} \) converges.
Problem 7.651medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{n + 2}{2 n + 2}\right)^{n} \) converges.
Problem 7.652medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{n + 3}{2 n + 1}\right)^{n} \) converges.
Problem 7.653medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{n + 3}{4 n + 1}\right)^{n} \) converges.
Problem 7.688medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{3 n}{4 n + 3}\right)^{n} \) converges.
Problem 7.689medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} 2^{- n} n^{4} \) converges.
Problem 7.690medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{4 n}{5 n + 1}\right)^{n} \) converges.
Problem 7.691medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{n}{n + 2}\right)^{n^{2}} \) converges.
Problem 7.692medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{4 n + 1}{5 n + 1}\right)^{n} \) converges.
Problem 7.693medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{3 n}{4 n + 1}\right)^{n} \) converges.
Problem 7.694medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{5 n + 1}{4 n + 2}\right)^{n} \) converges.
Problem 7.695medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{5 n}{4 n + 2}\right)^{n} \) converges.
Problem 7.696medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{3 n + 3}{2 n + 3}\right)^{n} \) converges.
Problem 7.722medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{2 n + 1}{n + 3}\right)^{n} \) converges.
Problem 7.723medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{4 n + 1}{3 n + 4}\right)^{n} \) converges.
Problem 7.724medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{3 n + 1}{2 n + 1}\right)^{n} \) converges.
Problem 7.725medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{3 n + 2}{4 n + 3}\right)^{n} \) converges.
Problem 7.726medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{5 n + 2}{4 n + 4}\right)^{n} \) converges.
Problem 7.727medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{4 n + 2}{3 n + 2}\right)^{n} \) converges.
Problem 7.728medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{5 n + 1}{4 n + 1}\right)^{n} \) converges.
Problem 7.729medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{4 n}{2 n + 4}\right)^{n} \) converges.
Problem 7.730medium✓ Every equation proved
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{5 n + 3}{n + 4}\right)^{n} \) converges.
Problem 7.756medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} 2^{- n} n \) converges.
Problem 7.757medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} 3^{- n} n^{3} \) converges.
Problem 7.758medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{3 n + 3}{2 n + 1}\right)^{n} \) converges.
Problem 7.759medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{5 n + 3}{n + 3}\right)^{n} \) converges.
Problem 7.760medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{5 n + 3}{2 n + 4}\right)^{n} \) converges.
Problem 7.761medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{3 n + 3}{4 n + 4}\right)^{n} \) converges.
Problem 7.762medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{3 n + 2}{2 n + 3}\right)^{n} \) converges.
Problem 7.763medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{2 n + 3}{n + 4}\right)^{n} \) converges.
Problem 7.764medium✓ Nihil obstat
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{4 n + 1}{2 n + 1}\right)^{n} \) converges.
Problem 7.765medium✓ Nihil obstat