Theory

Theory 1

A power series looks like this:

Power series are used to build and study functions. They allow a uniform “modeling framework” in which many functions can be described and compared. Power series are also convenient for computers because they provide a way to store and evaluate differentiable functions with numerical (approximate) values.

Small needed for power series

The most important fact about power series is that they work for small values of .

Many power series diverge for too big; but even when they converge, for big they converge more slowly, and partial sum approximations are less accurate.

The idea of a power series is a modification of the idea of a geometric series in which the common ratio becomes a variable , and each term has an additional coefficient parameter controlling the relative contribution of different orders.

Theory 2

Every power series has a radius of convergence and an interval of convergence.

Radius of convergence

Consider a power series centered at :

Apply the ratio test:

Define the radius of convergence :

Therefore:

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We can build shifted power series for near some other value . Just replace the variable with a shifted variable :

Now apply the ratio test to determine convergence:

Define the radius of convergence :

In the shifted setting, the radius of convergence limits the *distance from *:

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Method: To calculate the interval of convergence of a power series, follow these steps:

• Observe the center of the shifted series (or for no shift).
• Compute using the limit of coefficient ratios.
• Write down the preliminary interval .
• Plug each endpoint, and , into the original series
  • Check for convergence
• Add in the convergent endpoints. (4 possible scenarios.)