\( \newcommand{\sech}{\operatorname{sech}} \) \( \newcommand{\inverse}[1]{#1^\leftarrow} \) \( \newcommand{\<}{\langle} \) \( \newcommand{\>}{\rangle} \) \( \newcommand{\vect}{\mathbf} \) \( \newcommand{\veci}{\mathbf{\hat ı}} \) \( \newcommand{\vecj}{\mathbf{\hat ȷ}} \) \( \newcommand{\veck}{\mathbf{\hat k}} \) \( \newcommand{\curl}{\operatorname{curl}\,} \) \( \newcommand{\dv}{\operatorname{div}\,} \) \( \newcommand{\detThree}[9]{ \operatorname{det}\left( \begin{array}{c c c} #1 & #2 & #3 \\ #4 & #5 & #6 \\ #7 & #8 & #9 \end{array} \right) } \) \( \newcommand{\detTwo}[4]{ \operatorname{det}\left( \begin{array}{c c} #1 & #2 \\ #3 & #4 \end{array} \right) } \)

Section 5.8 Calculus 2

Strategies to Determine Series Convergence
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5.8 Strategies to Determine Series Convergence

5.8.1 Identifying Appropriate Strategies for Inspecting Series

  • When encountering a series, it’s useful to spot certain traits that can identify the best series convergence test to apply. The following list isn’t fool-proof, but checking these in order can help you identify useful techniques to determine series convergence.
    1. Is the value of a convergent series asked for? If so, it is likely a telescoping or geometric series.
    2. Is the sequence limit of its terms non-zero? If so, it diverges by the Series Divergence Test. (The opposite is NOT true.)
    3. Is the series of the form \(\sum ar^n\)? If so, it is a geometric series.
    4. Do the series terms alternate due to a multiplier of \((-1)^n\) or similar? If so, it is likely an alternating series.
    5. Is the series of the form \(\sum\frac{a}{n^p}\)? If so, it is a \(p\)-series.
    6. Does the series involve factorials? If so, the Ratio Test may be effective (but fails when the limit equals \(1\)).
    7. Does the series involve exponents? If so, the Root Test may be effective (but fails when the limit equals \(1\)).
    8. Can the series be slightly modified to a geometric or \(p\)-series? If so, then either the Direct or Limit Comparison Test may be effective.
  • Note that the starting index (the bottom of the \(\sum\)) does not affect whether a series converges or diverges (but it does affect the value of a convergent series).

Review Exercises

  1. Does \(\sum_{m=1}^\infty(-1)^{m-1}\frac{m+1}{2m}\) converge or diverge?
  2. Does \(\sum_{n=0}^\infty\frac{n+3}{3^n}\) converge or diverge?
  3. Does \(\sum_{k=0}^\infty 7^{1-k}\) converge or diverge? If it converges, what is its value?
  4. Does \(\sum_{j=4}^\infty\frac{5^j}{(2j)!}\) converge or diverge?
  5. Does \(\sum_{n=3}^\infty\frac{1}{n^2+n}\) converge or diverge? If it converges, what is its value?
  6. Does \(\sum_{k=1}^\infty(1+\frac{1}{k})^k\) converge or diverge?
  7. Does \(\sum_{m=2}^\infty5(-\frac{4}{9})^{m+1}\) converge or diverge?
  8. Does \(\sum_{n=0}^\infty\frac{n^3+n+7}{n^5+5n^2}\) converge or diverge?
  9. Does \(\sum_{k=10}^\infty\frac{4}{\sqrt[3]{k}}\) converge or diverge?
  10. For what values of \(x\) does \(\sum_{n=0}^\infty \frac{x^n}{n^2}\) converge?