Throughout these pages I will assume that you are familiar with power series and the concept of the radius of convergence of a power series.

Given a power series

we can find its derivative by differentiating term by term:

Here we used that the derivative of the term

The second derivative is computed similarly:

Taking the derivative of a power series does not change its radius of convergence, so will all have the same radius of convergence.

The rest of this section is devoted to "index shifting".
Consider the example

Using a simple substitution

or changing the dummy variable

The expression (

Summation is just a special case of integration, so an analogous "index shifting" will work:

You should convince yourself that both of these expressions are indeed the same, by writing out explicitly the four terms of each of the two formulas!

Let's try this for our derivative formulas:

We shifted

You should once again convince yourself that the first and the last formula are indeed the same, by writing out explicitly the first few terms of each of the two formulas!

As a last example, let's shift the formula for the second derivative by 2 units:

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