Well that went so well.

Let's do it again.

This time to estimate ARCSIN of one tenth within ten to the negative ten.

Now, I won't go through the details of the taylor expansion for ARCSIN of x.

The terms are a little bit complicated, but

not too bad if you assume that they're given.

What matters is the Taylor error bound that E sub N is less than a constant C

over (N+1) factorial times x to the N+1, where x equals one-tenth.

Now, this constant C is the critical piece of information.

It's an upper bound for the n plus first derivative

of ARCSIN(x) for all x between 0 and 1/10.

Now, who remembers the formula for the n plus first derivative of ARCSIN(x)?

Anybody?

I don't remember it either.

And this is the difficult part of using the Taylor bound.

You don't necessarily know a good bound for the N plus first derivative.

How are we going to solve this?

Well, if the Taylor theorem is not gonna work and it's not an alternating series,

and I don't think I wanna integrate this function, then what do we do?

Well, we're just going to have to think.

But, if we think, well this is not so bad.

Look at the terms in this series.

We have one tenth, and then something times one tenth cubed,

plus something times one tenth to the fifth, etc.

It seems as though every step where we go from n to n+2

we're picking up an extra one tenth squared.

Okay, so that's 1/100, but if we look at the coefficients the 2n+1 and

the product of odds over the products of evens,

then we're picking up another factor of 10 in the denominator.

And I claim that a, n+2 the next term in the series is less

than the previous term a sub n divided by 1,000.

What this means really is that you're picking up three decimal

places of accuracy, with each subsequent term.

And that means that if we want to get within 10 to the -10th,

it's going to suffice to choose N bigger than or equal to 7.

So the first four terms we have represented on this

slide suffice to approximate.

ARCSIN one-tenth within ten to the negative ten.

Never forget to think even if a Taylor bound doesn't work.

In general, bounding errors is just hard.

There's no getting around it.

If you're fortunate enough to have an alternating series, then it's not so bad.

If you've got something that works with an integral test, you're great.

If not, you're either going to have to resort to the Taylor theorem or

use your head.