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Wize University Calculus 2 Textbook > Sequences and Series

Which Convergence Test to Use?

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How to Pick Which Convergence Test to Use?

Try these in order:
1. Try simplifying the terms ana_n if possible.
2. Determine whether you have a common series
  • Geometric: ∑arn−1{\displaystyle \sum}ar^{n-1}
  • P-series: ∑1np{\displaystyle \sum}\frac{1}{n^p}
  • Harmonic series: ∑1n{\displaystyle \sum}\frac{1}{n} (special p-series where p=1)
3. Try the Test of Divergence (quickly check if lim⁡n→∞an≠0{\displaystyle \lim_{n\to\infty}}a_n\neq0)

4. If ana_n is a polynomial or rational function, try the Series Comparison Test or Limit Comparison Test
  • Compare it to a p-series (pick p by dividing the highest degree term in the numerator by the highest degree term in the denominator)
  • Sometimes you might compare it to a geometric series

5. If ana_n involves some form of (−1)n(-1)^n, the question might ask for absolute/conditional convergence. Try
  • Alternating Series Test (quickly check if bn+1≤bnb_{n+1}\le b_n and lim⁡n→∞bn=0{\displaystyle \lim_{n\to\infty}}b_n=0)
6. If the terms involve factorials or powers, try Ratio Test

7. If the terms look like f(n)g(n)f(n)^{g(n)}, try the Root Test

8. If ana_n looks like a function that can be integrated, try the integral test (check if the term function is positive, continous, and decreasing)


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What if the question asks us to find the value the series converges to?

  • Try to rewrite it into a geometric series (we know that ∑n=1∞arn−1=a1−r{\displaystyle \sum_{n=1}^\infty}ar^{n-1}=\frac{a}{1-r})
  • Try to expand out a few terms, you might get a telescoping sum (the middle terms all cancel out)
  • If it's a rational function that converges, try using partial fraction decomposition and then see if it's a telescoping sum
*This might also be a power series question