M344: Calculus III

Section 11.7

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Convergence Testing Strategies

By now we are familiar with all the different ways of testing a series for convergence or divergence, but we've studied each method in its own little basket of examples. The question becomes,

Given a series, which test should we use?

The following flow list can help us work through the process of deciding what test to use.

1. Test For Divergence
   1. Does $\lim\limits_{n\to\infty} a_n = 0$?
      1. No: The series $\sum a_n$ diverges.
      2. Yes: We don't know if the series $\sum a_n$ converges or diverges so we must use another test.

1. $p$-Series Test
   1. Does $a_n = \dfrac{1}{n^p}$, for $n \ge 1$?
      1. No: The series is not a $p$-series, use another test.
      2. Yes: Is $p>1$?
         1. No: The series $\sum a_n$ diverges.
         2. Yes: The series $\sum a_n$ converges.

1. Geometric Series Test
   1. Does $a_n = ar^{n-1}$ for $n\ge 1$?
      1. No: The series is not a geometric series, use another test.
      2. Yes: Is $\lvert r \rvert < 1$?
         1. No: The series $\sum a_n$ diverges.
         2. Yes: The series $\sum a_n$ converges and the sum is given by $\sum_{n=1}^{\infty} ar^{n-1} = \dfrac{a}{1 - r}$

1. Alternating Series Test
   1. Does $a_n = (-1)^{n}b_n$ or $a_n = (-1)^{n-1}b_n$, for $b_n \ge 0$?
      1. No: The series is not an alternating series, use another test.
      2. Yes: Is (1) $b_{n+1} \le b_n$ and (2) $\lim\limits_{n\to\infty} = 0$?
         1. No: Use another test.
         2. Yes: The series $\sum a_n$ converges.

1. Comparison Test
   1. Pick $\{b_n\}$. Does $\sum b_n$ converge?
      1. No: Is $0\le b_n \le a_n$?
         1. No: Choose another $b_n$ or use a different test.
         2. Yes: Since $b_n$ diverges AND $\le b_n \le a_n$, $a_n$ diverges.
      2. Yes: Is $0 \le a_n \le b_n$?
         1. No: Choose another $b_n$ or use a different test.
         2. Yes: Since $b_n$ converges AND $a_n \le b_n$, $a_n$ converges.

1. Limit Comparison Test
   1. Pick $\{b_n\}$. Does $\lim\limits_{n\to\infty}\dfrac{a_n}{b_n} = c > 0$, is $c$ is finite, AND are $a_n$, $b_n > 0$?
      1. No: Use a different test.
      2. Yes: Does $\sum_{n=1}^{\infty} b_n$ converge?
         1. No: The series $\sum a_n$ diverges.
         2. Yes: The series $\sum a_n$ converges.

1. Integral Test
   1. Can you write $a_n = f(n)$, so that $f(x)$ is continuous, positive, and decreasing on $[1, \infty)$
      1. No: Use a different test.
      2. Yes: Does $\int_{1}^{\infty} f(x) dx = c < \infty$ (i.e. converges)?
         1. No: The series $\sum a_n$ diverges.
         2. Yes: The series $\sum a_n$ converges.

1. Ratio Test
   1. Is $\lim\limits_{n\to\infty} \left\lvert \dfrac{a_{n+1}}{a_n} \right\lvert \ne 1$?
      1. No, it = 1: Use a different test (but not the Root Test).
      2. Yes: Is $\lim\limits_{n\to\infty} \left\lvert \dfrac{a_{n+1}}{a_n} \right\rvert < 1$?
         1. No: The series $\sum a_n$ diverges.
         2. Yes: The series $\sum a_n$ converges.

1. Root Test
   1. Is $\lim\limits_{n\to\infty} \sqrt[n]{\lvert a_n \rvert} \ne 1$?
      1. No, it = 1: Use a different test (but not the Ratio Test).
      2. Yes: Is $\lim\limits_{n\to\infty} \left\lvert \dfrac{a_{n+1}}{a_n} \right\rvert< 1$?
         1. No: The series $\sum a_n$ diverges.
         2. Yes: The series $\sum a_n$ converges.

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1. Example

Determine where the series is convergent or divergent.

$$ \sum_{n=1}^{\infty} \left( \dfrac{1}{n^3} + \dfrac{1}{3^n}\right) $$

Check your answer

The series is convergent.

Video solution

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2. Example

Determine whether the series converges or diverges.

$$ \sum_{n=1}^{\infty} \dfrac{n-1}{n^3+1} $$

Check your answer

The series is convergent.

Video solution

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3. Example

Determine whether the series converges or diverges.

$$ \sum_{n=1}^{\infty} \dfrac{\sqrt{n^4 + 1}}{n^3+n} $$

Check your answer

The series is convergent.

Video solution

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4. Example

Determine whether the series converges or diverges.

$$ \sum_{k=1}^{\infty} \dfrac{5^{k}}{3^{k}+4^{k}} $$

Check your answer

The series is divergent.

Video solution

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5. Example

Determine whether the series converges or diverges.

$$ \sum_{n=2}^{\infty} \dfrac{1}{\big(\ln(n)\big)^{\ln(n)}} $$

Check your answer

The series is convergent.

Video solution

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6. Example

Determine whether the series converges or diverges.

$$ \sum_{n=1}^{\infty} (-1)^n\dfrac{n^2-1}{n^2+1} $$

Check your answer

The series is divergent.

Video solution

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7. Example

Determine whether the series converges or diverges.

$$ \sum_{n=2}^{\infty} \dfrac{(-1)^{n-1}}{\sqrt{n} -1} $$

Check your answer

The series is convergent.

Video solution

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8. Example

Determine whether the series converges or diverges.

$$ \sum_{n=1}^{\infty} n^{2}e^{-n^{3}} $$

Check your answer

The series is convergent.

Video solution

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9. Example

Determine whether the series converges or diverges.

$$ \sum_{n=1}^{\infty} \left(\dfrac{n}{n+1}\right)^{n^{2}} $$

Check your answer

The series is convergent.

Video solution

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10. Example

Determine whether the series converges or diverges.

$$ \sum_{n=1}^{\infty} \dfrac{n!}{e^{n^{2}}} $$

Check your answer

The series is convergent.

Video solution

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Discussion

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