Section 10.1 Convergence and Values of Infinite Series
After defining series, in Subsection 9.2.3, I use partial sums to calculate the actual value of series. For geometric and telescoping series, I could produce actual values, usual rational numbers, for the sum of the series. Those were useful initial examples, but it turns out they were a bit misleading. For the vast majority of series, when the series does actual converge to a value, the value of the series is not going to be a familiar rational number (or even a familiar irrational numbers). Most series that do converge will converge to some new, unfamiliar number.
This reality changes the nature of the discussion of series. The naïve approach to series is to ask: What number does this series add up to?. That question assumes that the result will be familiar that series produce recognizable answers. If this is not the case, then I need to change the question to this: Is this series a number? That is, does the series actually converge.
In this spirit, what I care about is convergence, not calculation. If a series arises in some mathematical problem, what I most need to know is if it actually represents a number. I usually don’t need to know, at least at the start, roughly what the number is. I just need the confidence that there is a number, which I can then use for other purposes and calculation.
For this entire week, I’ll be introducing tests for convergence. These are various ways that I can determine whether a series converges: whether the I can treat it as a reasonable, well-defined number or not. I’m not going to actually try to calculate the values. The important matheamtics here is convergence, not calculation of values.
To try to make this point, conisder the following series.
\begin{equation*}
\sum_{n=1}^\infty \frac{n^2+4n}{(n!)2^n}
\end{equation*}
Using methods from later in this week, this series converges. This sum represents a number. Let me label that number.
\begin{equation*}
\alpha = \sum_{n=1}^\infty \frac{n^2+4n}{(n!)2^n}
\end{equation*}
Now I can do arithemtic with \(\alpha\text{.}\) I can use it in other equations, in other functions, in other series even. It is probably not a familiar number. I probably don’t have any other way to succinct write \(\alpha\) other than as this series. The series is the best way to refer to the number. The key is that I am assured that the series converges, that \(\alpha\) is some number.
Though the focus of this week is on convergence, eventually, I might like to know the rough value of the number. Though I can’t likely ‘calculate’ the series by writing \(\alpha\) as some familiar number, I can use the series to approximate the number. For example, here is the fourth partial sum of the series.
\begin{equation*}
\frac{5}{2} + \frac{12}{8} + \frac{21}{48} + \frac{32}{384} =
\frac{217}{48}
\end{equation*}
This is an approximation for \(\alpha\text{.}\) If I add more terms, I will get a closer approximation. This is the best way to get a representation of \(\alpha\) in familiar terms. I could wonder how accurate this approximation is. The accuracy and precision of series approximation is an important question in the study of infinite series and itself is a bit part of mathematics. In this course, I’ll not cover much about accuracy of approximation with series. However, this will be covered in other courses and is particular important for numerical analysis and algorithm analysis in computing science. Indeed, all the work that computers and calculators do to calculate values of functions is approximation work, which requires careful control of the error to ensure that answers are given to the expected precision.
