Intermediate Integration

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Using the Constant Multiple Rule for Simplifying a Series

The Constant Multiple Rule for integration allows you to simplify an integral by factoring out a constant. This option is also available when you’re working with series. Here’s the rule: [more…]

Using the Sum Rule for Simplifying a Series

The Sum Rule for integration allows you to split a sum inside an integral into the sum of two separate integrals. Similarly, you can break a sum inside a series into the sum of two separate series: [more…]

Understanding Power Series

The geometric series is a simplified form of a larger set of series called the power series. A power series is any series of the following form: [more…]

Understanding the Interval of Convergence

Unlike geometric series and p-series, a power series often converges or diverges based on its x value. This leads to a new concept when dealing with power series: the interval of convergence. [more…]

Expressing Functions as Power Series Using the Maclaurin Series

The Maclaurin series is a template that allows you to express many other functions as power series. It is the source of formulas for expressing both sin [more…]

Expressing Functions as Power Series Using the Taylor Series

The Taylor series provides a template for representing a wide variety of functions as power series. It is relatively simple to work with, and you can tailor it to obtain a good approximation of many functions [more…]

Determining Whether a Taylor Series Is Convergent or Divergent

Because the Taylor series is a form of power series, every Taylor series also has an interval of convergence. When this interval is the entire set of real numbers, you can use the series to find the value [more…]

Calculating Error Bounds for Taylor Polynomials

A Taylor polynomial approximates the value of a function, and in many cases, it’s helpful to measure the accuracy of an approximation. This information is provided by the [more…]

Determine Signed Areas in a Problem

The solution to a definite integral gives you the signedarea of a region. In some cases, signed area is what you want, but in some problems you’re looking for [more…]

Determine Unsigned Area between Curves

You can use the concept of unsigned area to measure the area between curves. For example, you can use this technique to find the unsigned shaded area in the following figure. [more…]

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