5.6 Ratio and Root Tests

Ratio and root tests are powerful tools for determining if infinite series converge or diverge. They work by analyzing how quickly terms approach zero as the series progresses.

These tests are particularly useful for series with factorials or exponential terms. When one test fails, the other might succeed, making them complementary methods for tackling tricky convergence problems.

Ratio and Root Tests for Series Convergence

Ratio test for series convergence

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• Determines convergence or divergence of infinite series by analyzing the ratio of successive terms
  • Considers series in the form $\sum_{n=1}^{\infty} a_n$
  • Computes limit of absolute value of ratio between consecutive terms: $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L$
• Series is absolutely convergent when $L < 1$ (terms decrease rapidly)
• Series is divergent when $L > 1$ (terms grow or decrease slowly)
• Ratio test is inconclusive when $L = 1$, requiring other convergence tests
• Particularly useful for series involving factorials (n!) or exponential terms ($e^n$)
• Can be used to determine the radius of convergence for power series

Root test for series convergence

• Alternative method to determine convergence or divergence of infinite series
  • Considers series in the form $\sum_{n=1}^{\infty} a_n$
  • Computes limit of nth root of absolute value of nth term: $\lim_{n \to \infty} \sqrt[n]{|a_n|} = L$
• Series is absolutely convergent when $L < 1$ (terms approach 0 rapidly)
• Series is divergent when $L > 1$ (terms approach 0 slowly or diverge)
• Root test is inconclusive when $L = 1$, requiring other convergence tests
• Often used when ratio test fails or is difficult to apply (complex ratios)

Systematic approach to convergence tests

• Step-by-step process to determine convergence or divergence of infinite series:
  1. Identify known series types (geometric series with $|r| < 1$, p-series with $p > 1$)
  2. Apply divergence test: if $\lim_{n \to \infty} a_n \neq 0$, series diverges (terms don't approach 0)
  3. Check for alternating series and apply alternating series test if applicable (terms alternate signs and decrease in absolute value)
  4. Apply ratio test or root test if suitable (based on series form)
  5. If ratio or root test is inconclusive, use other tests:
     • Comparison test (compare with known convergent or divergent series)
     • Limit comparison test (analyze limit of ratio between series and known series)
     • Integral test (compare series with improper integral)
• Understand conditions and limitations of each test (applicable series forms)
• Practice applying various tests to different series (polynomial, exponential, logarithmic) to strengthen understanding of test usage