The Unapologetic Mathematician

Infinite Series

And now we come to one of the most hated parts of second-semester calculus: infinite series. An infinite series is just the sum of a (countably) infinite number of terms, and we usually collect those terms together as the image of a sequence. That is, given a sequence $a_k$ of real numbers, we define the sequence of “partial sums”:

$\displaystyle s_n=\sum\limits_{k=0}^na_k$

and then define the sum of the series as the limit of this sequence:

$\displaystyle\sum\limits_{k=0}^\infty a_k=\lim\limits_{n\rightarrow\infty}s_n$

Notice, though, that we’ve seen a way to get finite sums before: using step functions as integrators. So let’s use the step function $\lfloor x\rfloor$, which is defined for any real number $x$ as the largest integer less than or equal to $x$.

This function has jumps of unit size at each integer, and is continuous from the right at the jumps. Further, over any finite interval, its total variation is finite. Thus if $f$ is any function continuous from the left at every integer it will be integrable with respect to $\lfloor x\rfloor$ over any finite interval. Further, we can easily see

$\displaystyle\int\limits_a^bf(x)d\lfloor x\rfloor=\sum\limits_{\substack{k\in\mathbb{Z}\\a<k\leq b}}f(k)$

Now given any sequence $a_n$ we can define a function $f$ by setting $f(x)=a_{\lceil x\rceil}$ for any $x>-1$. That is, we round each number up to the nearest integer $n$ and then give the value $a_n$. This gives us a step function with the value $a_n$ on the subinterval $\left(n-1,n\right]$, which we see is continuous from the left at each jump. Thus we can always define the integral

$\displaystyle\int\limits_{-\frac{1}{2}}^bf(x)d\lfloor x\rfloor=\sum\limits_{k=0}^{\lfloor b\rfloor}a_k=s_{\lfloor b\rfloor}$

Then as we let $b$ go to infinity, $\lfloor b\rfloor$ goes to infinity with it. Thus the sum of the series is the same as the improper integral.

So this shows that any infinite series can be thought of as a Riemann-Stieltjes integral of an appropriate function. Of course, in many cases the terms $a_k$ of the sequence are already given as values $f(k)$ of some function, and in that case we can just use that function instead of this step-function we’ve cobbled together.

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Posted by John Armstrong | Analysis, Calculus

8 Comments	»

1. [...] Tests for Infinite Series Now that we’ve seen infinite series as improper integrals, we can immediately import our convergence tests and apply them in this special [...]

Pingback by | April 25, 2008 | Reply

2. [...] can forge a direct connection between the sum of an infinite series and the improper integral of a function using the famed integral test for [...]

Pingback by | April 29, 2008 | Reply

3. [...] Partial Summation Formula When we consider an infinite series we construct the sequence of partial sums of the series. This is something like the indefinite [...]

Pingback by | April 30, 2008 | Reply

4. [...] of longer and longer sequences of terms and talk sensibly about whether these sums — these series — converge or not. Unfortunately, this topological concept ends up breaking the algebraic [...]

Pingback by | May 6, 2008 | Reply

5. [...] we’re back in the realm of infinite series, and taking the limit of a sequence of partial sums. The series in question has as its th term the [...]

Pingback by | August 27, 2008 | Reply

6. [...] of Complex Series Today, I want to note that all of our work on convergence of infinite series carries over — with slight modifications — to complex [...]

Pingback by | August 28, 2008 | Reply

7. [...] long as this infinite series converges, and define it to be if the series diverges. We then define the outer Lebesgue measure [...]

Pingback by | December 10, 2009 | Reply

8. [...] I dealt with what happened when we wanted to integrate over an infinite interval. Then I defined an infinite series to be what happened when we used a particular integrator in our Riemann-Stieltjes integral. This [...]

Pingback by | January 15, 2010 | Reply

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