# How do you use a Power Series to estimate the integral #int_0^0.01cos(sqrt(x))dx# ?

Since

we have

Now, consider the integral in question.

by integrating term by term,

By adding a few terms of the above series, we can approximate the value of the original definite integral.

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To estimate the integral (\int_{0}^{0.01} \cos(\sqrt{x}) , dx) using a power series, we first need to represent (\cos(\sqrt{x})) as a power series. One way to do this is by using the Maclaurin series expansion of (\cos(x)) and then substituting (x) with (\sqrt{x}).

The Maclaurin series expansion of (\cos(x)) is:

[ \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} ]

Substituting (x) with (\sqrt{x}), we get:

[ \cos(\sqrt{x}) = \sum_{n=0}^{\infty} \frac{(-1)^n x^n}{(2n)!} ]

Then, we integrate this power series term by term from (0) to (0.01):

[ \int_{0}^{0.01} \cos(\sqrt{x}) , dx = \int_{0}^{0.01} \sum_{n=0}^{\infty} \frac{(-1)^n x^n}{(2n)!} , dx ]

Because the series representation is an infinite sum, integrating term by term can be computationally intensive. However, we can truncate the series to a finite number of terms based on the desired level of accuracy.

After integrating, we can compute the sum of the series for the given limits to estimate the value of the integral.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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