Let's not start with the zero of term.
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How do I sum this geometric series? It's the sum
k goes from m to infinity of r to the k.
So this starts out with a term r to the m and then the next
term is r to the m plus 1, and it keeps on going like that.
It's a geometric series but the first term is r to the m, not r to the 0.
I remember how to sum the series that starts not with m but with 0.
Well, in that case, to evaluate the sum k goes from 0 to infinity of r to the k.
We already studied that, and that's 1 over 1 minus r,
provided the absolute value of r is less than 1.
I want to somehow change that 0 back into an m.
there's a general result that I can invoke.
Some constant times the series, which adds up all the a sub
k's is equal to the series that adds up c times
the a sub k's.
This is of course assuming that the series makes sense.
Right?
I'm assuming that this series converges in order to assert this equality.
So, in this case, I'll multiply by r to the m.
So let me write that down.
I'm going to multiply both sides by r to the m.
So the r to the m times the sum k goes from 0 to infinity
of r to the k is r to the m over 1 minus r.
Now I can use this result.
Remember that this result is assuming that this series converges.
So I'm always going to be working with the assumption that the absolute value
of r is less than 1 in order to guarantee that this series converges.
Oh, wait, but yeah, let's apply the result.
So if I apply the result in this case, what do I get?
Well, here, r to the m is playing the role of c,
and r to the k is a sub k. So this can be replaced
by this. Which means this is equal to what?
It's equal to the sum k goes from 0 to infinity of
r to the m times r to the k. Which I could rewrite as the sum
k goes from 0 to infinity of r to the m plus k.
The bounds on the series can be rewritten. Well, what do I mean by that?
Well, let me write out what this series looks like.
I'm just going to write out the first few terms of the series.
The k equals 0 term is
r to the m. The k equals 1 term is r to to
the m plus 1. The k equals 2 term is r
to the m plus 2, and so on. So, I could
rewrite this series as the series that I'm interested in.
This series ends up just being the sum k goes from m to infinity of r to the k.
And if I go all the way back, right. I figured out that that series
has this value of course assuming that the absolute value of r is less than 1.
But all told then, I can conclude that this series
is equal to r to the m over 1 minus
r, as long as the absolute value of r is less than 1.
So that's a useful formula, but even more useful than
the formula is the method that we used for deriving it.
We used this fact to derive that formula.
But this fact, while it looks like the
distributive law is more than just the distributive law.
What I mean is that the distributive law says this.
It says that C times a sub 0 plus a sub 1 plus dot, dot, dot plus a sub n
is equal to C times a sub 0 plus C times a
sub 1 plus dot, dot, dot plus C times a sub n.
It's something about finite sums.
I could even write it using summation notation.
The distributive law says that C times the sum k goes from 0 to n of
a sub k is equal to the sum of k goes from 0 to n of C times a
sub k.
That's what the distributive law is telling me.
But contrast that with this statement.
All right?
This isn't just a statement about finite
sums, this is a statement about infinite series.
So how do I justify something like that?
Well, it's more than just the distributive law.
Let me show you.
So, I could write down C times the limit as n approaches infinity of the sum
k goes from 0 to n of a sub k. And by definition, this is C
times the infinite series k goes from 0 to infinity of a sub k.
But I know something about limits.
A constant multiple of a limit is the limit of
that constant multiple, times whatever I'm taking the limit of.
Right?
So this thing here is equal to the limit as
n approaches infinity of C times the sum k goes from 0 to n of a sub k.
And now I can apply the distributive law because this is just a finite sum.
It's a finite sum where n is maybe very big.
But it still is a finite sum.
So this is the limit n goes to infinity of the sum k goes from
0 to n of C times a sub k. So this equality is the distributive law.
Now I've got the limit of this sum, and that is the infinite series k goes
from 0 to infinity of C times a sub k. So let's look at what happened here.
All right?
This is just the definition of the infinite series.
The fact that these are equal is a property about limits.
The fact that these are equal is the distributive law.
And the fact that these are equal is the definition of infinite series.
So the fact that a constant multiple times an infinite series is the infinite series
of that constant multiple of the terms is
more than just the distributive law at play.
It's really the distributive law plus a fact about limits.
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