The Unapologetic Mathematician

Mathematics for the interested outsider

Examples of Convergent Series

Today I want to give two examples of convergent series that turn out to be extremely useful for comparisons.

First we have the geometric series whose terms are the sequence a_n=a_0r^n for some constant ratio r. The sequence of partial sums is

\displaystyle\sum\limits_{k=0}^na_0r^k=a_0\left(1+r+r^2+...+r^n\right)

If r\neq1 we can multiply this sum by \frac{1-r}{1-r} to find

\displaystyle\sum\limits_{k=0}^na_0r^k=a_0\frac{1-r^{n+1}}{1-r}

Then as n goes to infinity, this sequence either blows up (for |r|>1) or converges to \frac{a_0}{1-r} (for |r|<1). In the border case r=\pm1 we can also see that the sequence of partial sums fails to converge. Thus the geometric series converges if and only if |r|<1, and we have a nice simple formula telling us the sum.

The other one I want to hit is the so-called p-series, whose terms are a_n=n^{-p} starting at n=1. Here we use the integral test to see that

\displaystyle\lim\limits_{n\rightarrow\infty}\left(\sum\limits_{k=1}^n\frac{1}{n^p}-\int\limits_1^n\frac{dx}{x^p}\right)=D

so the sum and integral either converge or diverge together. If p\neq1 the integral gives \frac{n^{1-p}-1}{1-p}, which converges for p>1 and diverges for p<1.

If p=1 we get \ln(n), which diverges. In this case, though, we have a special name for the limit of the difference D. We call it “Euler’s constant”, and denote it by \gamma. That is, we can write

\displaystyle\sum\limits_{k=1}^n\frac{1}{k}=\ln(n)+\gamma+e(n)

where e(n) is an error term whose magnitude is bounded by \frac{1}{n}.

In general we have no good value for the sums of these series, even where they converge. It takes a bit of doing to find \sum\frac{1}{n^2}=\frac{\pi^2}{6}, as Euler did in 1735 (solving the “Basel Problem” that had stood for almost a century), and now we have values for other even natural number values of p. The sum \sum\frac{1}{n^3} is known as Apéry’s constant, after Roger Apéry who showed that it was irrational in 1979. Yes, we didn’t even know whether it was a rational number or not until 30 years ago. We have basically nothing about odd integer values of p.

If we say s instead of p, and let s take complex values (no, I haven’t talked about complex numbers yet, but some of you know what they are) we get Riemann’s function \zeta(s)=\sum\frac{1}{n^s}, which is connected to some of the deepest outstanding questions in mathematics today.

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April 29, 2008 - Posted by | Analysis, Calculus

4 Comments »

  1. [...] if we set , this tells us that . Then the comparison test with the geometric series tells us that [...]

    Pingback by The Ratio and Root Tests « The Unapologetic Mathematician | May 5, 2008 | Reply

  2. [...] the complex norm is multiplicative, everything for the geometric series goes through again: if , and it diverges if . The case where is more complicated, but it can be [...]

    Pingback by Convergence of Complex Series « The Unapologetic Mathematician | August 28, 2008 | Reply

  3. [...] the final summation converges because it’s a geometric series with initial term and ratio . This implies that [...]

    Pingback by Some Sets of Measure Zero « The Unapologetic Mathematician | December 14, 2009 | Reply

  4. [...] this is a chunk of a geometric series; since , the series must converge, and so we can make this sum as small as we please by choosing [...]

    Pingback by The Picard Iteration Converges « The Unapologetic Mathematician | May 6, 2011 | Reply


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