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10. Convergence of Series

≪ 9. Improper Integrals | Table of Contents

One of the hallmarks of calculus is the theory of infinite series, particularly that of Taylor series and power series. This is a framework in which certain “nice” functions are expressed as f(x)=n=0anxnf(x) = \sum _{n=0}^{\infty} a_n x^n for a sequence of real numbers ana_n, for instance. From a practical standpoint, this is particularly useful when computing values of certain functions, and many computer math libraries and calculators do this under the hood. Power series can, in certain scenarios, be much easier to manipulate than arbitrary functions, aiding both theory and applications.

In these scenarios, it’s important for people to know (1) for which values of xx does the series converge to an actual number, and (2) how many terms from the infinite series need to be added up in order to get “close enough” to the actual value. Our focus at the tail end of this class will be the first question — how can you tell if a series converges or not?

If n=0an\sum _{n=0}^{\infty} a_n converges, then necessarily limnan=0\lim _{n\to\infty} a_n = 0. If one wants to add up infinitely many numbers, those numbers must be getting smaller; this is called the divergence test. However, there are still sequences with limnan=0\lim _{n\to\infty}a_n = 0 for which the series still diverges — the most famous example is the harmonic series n=11n=\sum _{n=1}^{\infty} \frac{1}{n} = \infty. We have to use more sophisticated reasoning in these cases.

Most of the time, it’ll be with a comparison test — there are a small handful of “baseline” series that have been known to converge or diverge since antiquity.

These are facts that should be burned into your mind. I think the geometric series makes sense — at the very least, if r1\left\lvert r \right\rvert \geq 1, then you’re adding infinitely many things that aren’t getting any smaller, and there’s no way that should be a finite quantity. The pp-series should be reminiscent of the improper integrals 11xpdx\int _{1}^{\infty} \frac{1}{x^p} dx, which converge with exactly the same conditions on pp. In fact, we’re doing an “improper Riemann sum”.

Not every series is of these forms, but you should think of using these as “references”. Does my series n=0an\sum _{n=0}^{\infty} a_n resemble a geometric series in a quantitative way? Does it resemble a pp-series in a quantitative way? This is what convergence tests do.

Our first pair of convergence tests assess how similar a given series is to a geometric series.

Theorem 1. The Ratio Test

Let ana_n be a sequence of real numbers, and suppose the limit L=limnan+1an.L = \lim _{n\to\infty} \frac{\left\lvert a _{n+1} \right\rvert}{\left\lvert a_n \right\rvert}. Then n=0an\sum _{n=0}^{\infty}a_n

  • converges if L<1L < 1,
  • diverges if L>1L > 1, and
  • the test is inconclusive if L=1L = 1.

If the ratio of consecutive terms is LL, then eventually the terms start looking like part of a geometric series. Each term is (more or less) a factor of LL away from the previous. Thus, the cases L>1L> 1 and L<1L< 1 reflect exactly the convergence and divergence of geometric series. Do note that when L=1L = 1, it’s impossible to tell if a series converges or diverges using this test alone; you’ll need to use one of the other tests to find out.

The ratio test is particularly useful when there are powers of a constant and/or factorials involved, as these cancel nicely in the ratios.

Example 2.

Determine if the series n=05nn!\sum _{n=0}^{\infty} \frac{5^n}{n!} converges or diverges.

Solution

Let an=5nn!a_n = \frac{5^n}{n!}. Then, limnan+1an=limn5n+1(n+1)!n!5n=limn5n+1=0.\lim _{n\to\infty} \left\lvert \frac{a _{n+1}}{ a_n } \right\rvert= \lim _{n\to\infty} \frac{5 ^{n+1}}{(n+1)!} \cdot \frac{n!}{5^n} = \lim _{n\to\infty} \frac{5}{n+1}=0. Note here that (n+1)!=12n(n+1)=n!(n+1)(n+1)! = 1\cdot 2 \cdots n \cdot (n+1) = n! \cdot (n+1).

Since 0<10 < 1, the series converges by the ratio test.

Example 3.

Determine if the series n=1n!nn\sum _{n=1}^{\infty} \frac{n!}{n^n} converges or diverges.

Solution
Again, we’ll use the ratio test. Let an=n!nna_n = \frac{n!}{n^n}. We have L=limnan+1an=limn(n+1)!(n+1)n+1nnn!.L = \lim _{n\to\infty} \left\lvert \frac{a _{n+1}}{a_n} \right\rvert = \lim _{n\to\infty} \frac{(n+1)!}{(n+1) ^{n+1}} \cdot \frac{n^n}{n!}. Since (n+1)!=(n+1)n!(n+1)! = (n+1) \cdot n!, just like the last problem, the n!n!’s cancel each other out. The remaining n+1n+1 cancels out with one power of (n+1)n+1\left( n+1 \right) ^{n+1}, leaving just (n+1)n(n+1)^n. Thus, we get L=limnnn(n+1)n=limn(nn+1)n.L = \lim _{n\to\infty} \frac{n^n}{(n+1)^n} = \lim _{n\to\infty} \left( \frac{n}{n+1} \right)^n. We have to use L’Hôpital’s rule to evaluate this limit! (Take natural logs of everything.) You’ll eventually end up with L=1e<1,L = \frac{1}{e} < 1, hence this series again converges by the ratio test.

There’s another test that works on a similar principle to the ratio test, but makes the connection to geometric series in a slightly different way.

Theorem 4. The Root Test

Let ana_n be a sequence, and let L=limnan1n.L = \lim _{n\to\infty} \left\lvert a_n \right\rvert ^{\frac{1}{n}}. Then, the series n=1an\sum _{n=1}^{\infty} a_n

  • converges if L<1L < 1,
  • diverges if L>1L > 1, and
  • the test is inconclusive if L=1L = 1.

Just like the ratio test, the root test uses the heuristic that anLna_n \approx L^n when nn is very, very large. Thus, the conditions for convergence and divergence when L1L \neq 1 again mimick those of normal geometric series, though again one can’t use this test when L=1L = 1. This test is particularly useful when there are lots of things being raised to the nn-th power, but it’s a lot harder to apply when factorials or polynomial quantities are present.

A more direct set of convergence tests are the comparison tests, and these are often important to use when the ratio test or the root test are inconclusive.

Theorem 5.

Let ana_n and bnb_n be two sequences.

  1. (Direct comparison)
    If anbn\left\lvert a_n \right\rvert \leq b_n for all nn and n=0bn\sum _{n=0}^{\infty} b_n converges, then n=0an\sum _{n=0}^{\infty} a_n converges too. If anbn>0a_n\geq b_n > 0 and n=0bn\sum _{n=0}^{\infty} b_n diverges, then so does n=0an\sum _{n=0}^{\infty} a_n.
  2. (Limit comparison)
    If limnanbn\lim _{n\to\infty} \left\lvert \frac{a_n}{b_n} \right\rvert exists and is nonzero, then n=0an\sum _{n=0}^{\infty} a_n converges if n=0bn\sum _{n=0}^{\infty} b_n converges, and it n=0an\sum _{n=0}^{\infty} a_n diverges if n=0bn\sum _{n=0}^{\infty} b_n diverges.

The direct comparison test says (morally) that if n=0bn<\sum _{n=0}^{\infty} b_n < \infty, then n=0ann=0ann=0bn<.\left\lvert \sum _{n=0}^{\infty} a_n \right\rvert \leq \sum _{n=0}^{\infty} \left\lvert a_n \right\rvert \leq \sum _{n=0}^{\infty} b_n < \infty. Thus the series for ana_n is finite, i.e. it converges. Likewise, if you can bound ana_n from below by bnb_n and n=0bn\sum _{n=0}^{\infty} b_n diverges, then n=0an\sum _{n=0}^{\infty} a_n is “bigger than \infty”, hence must also diverge. Note that the other comparisons do not work in our favour.

The limit comparison test instead says that if ana_n eventually looks like a constant multiple of the sequence bnb_n, then the two series must both converge or both diverge.

These tests are most useful when the ratio and root tests are inconclusive, particularly when the summand is close to a rational function of nn. You are usually applying these tests to determine if n=0an\sum _{n=0}^{\infty} a_n converges or not, where ana_n is given as part of a problem. You have complete freedom to choose bnb_n, and almost always you’ll want to pick a geometric series or a pp-series, whose convergence is easy to assess.

Example 6.

Determine if the series n=1n+43n3+2n+1\sum _{n=1}^{\infty} \frac{n+4}{3n^3+2n+1} converges or diverges.

Solution

For problems like this, I would recommend squinting your eyes and asking, when nn is really big, what’s the simplest approximation I can make?

When nn is say 101010 ^{10}, we have n+4nn+4 \approx n and 3n3+2n+13n33n^3+2n+1 \approx 3n^3. Thus, I would like to say n=1n+43n3+2n+1n=11n2,\sum _{n=1}^{\infty} \frac{n+4}{3n^3+2n+1} \approx \sum _{n=1}^{\infty} \frac{1}{n^2}, and I know this is a convergent pp-series. To make this rigorous, I might apply the limit comparison test: if an=n+43n3+2n+1a_n = \frac{n+4}{3n^3+2n+1} and bn=1n2b_n = \frac{1}{n^2}, we have limnanbn=limnn3+4n23n3+2n+1=13.\lim _{n\to\infty} \frac{a_n}{b_n} = \lim _{n\to\infty} \frac{n^3+4n^2}{3n^3+2n+1} = \frac{1}{3}. This is a finite, nonzero quantity, and since n=1bn<\sum _{n=1}^{\infty} b_n <\infty, the series n=1an\sum _{n=1}^{\infty} a_n converges too.

Example 7.

Determine if the series n=12n+3n14n+lnn+2n\sum _{n=1}^{\infty} \frac{2n+3^n}{14\sqrt n + \ln n + 2^n} converges or diverges.

Solution

Although this looks more intimidating, the exact same thought process from the previous example applies. Note that we can’t immediately apply the ratio or the root test — there’s a lot of extra junk in the way that would prevent easy simplification.

Instead, when nn is really big, 2n+3n3n2n+3^n\approx 3^n while 14n+lnn+2n2n14\sqrt n + \ln n + 2^n \approx 2^n. So, I would like to say that n=12n+3n14n+lnn+2nn=1(32)n,\sum _{n=1}^{\infty} \frac{2n+3^n}{ 14\sqrt n + \ln n + 2^n} \approx \sum _{n=1}^{\infty} \left( \frac{3}{2} \right)^n, and I know the rightmost side is a diverging geometric series.

Rather than wrangle with the limit comparison test, let’s demonstrate how to use the direct comparison test. I need to make 2n+3n14n+lnn+2nC3n2n,\frac{2n+3^n}{14\sqrt n + \ln n + 2^n} \geq C \cdot \frac{3^n}{2^n}, where CC can be any constant. Note that I used the “squint your eyes” test to determine which direction I need the inequality to go.

In the numerator, I trivially have 2n+3n3n2n+3^n \geq 3^n for all nn. In the denominator, I have that n2n\sqrt n \leq 2^n and lnn2n\ln n \leq 2^n for all nn, thus 14n+lnn+2n162n14 \sqrt n + \ln n + 2^n \leq 16 \cdot 2^n for all nn. Note that the inequality is reversed! Putting the two together, we get 2n+3n14n+lnn+2n3n162n.\frac{2n+3^n}{14\sqrt n + \ln n + 2^n} \geq \frac{3^n}{16\cdot 2^n}. Since the series n=03n162n\sum _{n=0}^{\infty} \frac{3^n}{16 \cdot 2^n} is a diverging geometric series, we know n=12n+3n14n+lnn+2n\sum _{n=1}^{\infty} \frac{2n+3^n}{14\sqrt n + \ln n + 2^n} diverges too by the comparison test.