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Let's start with something big, something fundamental.

The Fundamental Theorem of Integral Calculus has a digital form.

It takes the form of the sum, as n goes from A to B

and the forward difference of a sequence U is equal to that

sequence U. Evaluated from N equals A to N equals

B plus one. Now what does that mean?

That means you evaluate the anti-derivative

that is the anti-difference at the end

points, but notice, that there's a, a little bit of a difference here.

We have to add a plus one to the upper limit.

That's in a case of a forward

difference, for backward difference, there is one subtracted

from the lower limit.

Now what's the proof of the Fundamental Theorem of discrete calculus.

Well, let's move this sum up, and expand out exactly what it means.

Remember, the forward difference says, we take the

next term in the sequence, minus the current term.

So if we add up all of the forward differences

as n goes from A to B. Then what do we see?

We see that most of the terms cancel,

because you have a plus, and then a minus of the same term.

The only things that do not cancel are the first term, negative

U sub A and the last term, plus U sub B plus 1.

And that is, indeed, as we have written it, the sequence evaluated at the limits.

Now, sometimes, these kinds of sums are called Telescoping Sums.

But if you ever see reference to that, now

you know you're really seeing the fundamental theorem in disguise.

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Here are a few examples of the fundamental theorem of discrete calculus in action.

Since the forward difference of the sequence 1 over n is equal to, by

definition 1 over n plus 1, minus 1 over n, which when

simplified is negative 1 over n squared plus n, then we can,

if you like, integrate both sides. To obtain, an explicit

formula for the sum as n goes from A to B of one over

n squared plus n. It is the anti-derivative, if you will.

Negative one over n evaluated from A to B plus one.

This can be written with a little bit of algebraic simplification.

As B minus A plus 1 over A times quantity B plus 1.

Here's another example.

Since the forward difference of the sequence n factorial

is, by definition, n plus one factorial minus n factorial.

Which after factoring out n factorial gives n factorial times n,

then we get an unusual looking formula. The sum, as n goes from A to B of n

factorial times n, is equal to n factorial, evaluated

from A to B plus one. That's simply B plus one factorial

minus A factorial, very, very simple.

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Now, since we know a few things about the

differences of falling powers, we can say some stronger things.

For example, the sum as n goes from one to k of n.

We've seen that sum before.

Here's a way to get it by means of integration.

This is really the sum of n to the falling one power.

What's the anti-difference

of that? Well, it's one half n to the falling two .

When we evaluate that from one to k plus one, we get k plus one times k over two.

Now, you've seen this formula before.

But I bet you've never seen this simple derivation.