- Once we have some series in hand, we might want to start operating with them. In particular, as we saw in our very first series, we might want to add up the terms:
In particular, we expect that
Such as sum is called an infinite series.
- The question which presents itself is this: what do we mean by this sum of an infinite number of terms? This is indeed curious.
- Here's how we make sense of that infinite sum:
- From the sequence
we define a new sequence:
These are called partial sums. If
Then we say that
exists, and is equal to L. More formally,
Now geometric series are sufficiently important that it's useful to include that special case:
There's a really fun proof of the first part above. Let's have a look at it.
This theorem tells us that if a sequence isn't asymptotic to the x-axis, we can forget about its partial sums converging.
Finally, series behave the way we'd hope:
- Examples:
- Reflect back on #36, p. 555. Recursive actions and definitions give rise to these types of geometric sums.
- Along the same lines: #45, p. 555
- #7, p. 554
- #8, p. 554 (compare to Taylor series error bound)
- #52, p. 556
- Xlispstat file:
- From the sequence