The Root Test is a convergence test for infinite series that is especially useful when the nth term contains powers such as n, 2n, or n squared in an exponent. It asks how large the typical term is after taking the nth root, which reveals the term's long-term exponential behavior. This matters because exponential growth or decay usually determines convergence faster than polynomial factors.
The test gives a clear three-way decision: converge, diverge, or inconclusive.
Key Facts
- For a series sum a_n, compute L = lim as n approaches infinity of nth root of |a_n|.
- If L < 1, then sum a_n converges absolutely.
- If L > 1 or L = infinity, then sum a_n diverges.
- If L = 1, the Root Test is inconclusive.
- The Root Test is strongest when a_n has the form (expression)^n or contains powers depending on n.
- Useful limit: lim as n approaches infinity of n^(1/n) = 1.
Vocabulary
- Infinite series
- An infinite series is a sum of terms a_1 + a_2 + a_3 + ... that continues without end.
- Convergence
- Convergence means the partial sums of an infinite series approach a finite number.
- Absolute convergence
- A series sum a_n converges absolutely if the series sum |a_n| converges.
- nth root
- The nth root of a number x is the value that becomes x when raised to the nth power.
- Inconclusive test
- An inconclusive test result means the test does not decide whether the series converges or diverges.
Common Mistakes to Avoid
- Forgetting the absolute value in nth root of |a_n| is wrong because the Root Test checks absolute convergence and must handle negative or alternating terms.
- Concluding convergence when L = 1 is wrong because the Root Test gives no information in that case and another test is needed.
- Applying the Root Test to the sequence a_n instead of the series sum a_n is wrong because convergence is about the sum of infinitely many terms, not just the term formula alone.
- Ignoring dominant exponential factors is wrong because the Root Test is designed to detect exponential behavior, while slower polynomial factors often disappear after taking the nth root.
Practice Questions
- 1 Use the Root Test to determine whether the series sum from n = 1 to infinity of (3n/(5n + 1))^n converges or diverges.
- 2 Use the Root Test to determine whether the series sum from n = 1 to infinity of n^2/4^n converges or diverges.
- 3 Explain why the Root Test is often more efficient than the Ratio Test for a series whose nth term is ((n + 2)/(3n - 1))^n.