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How to write functions as power series

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We use the polynomial with infinitely many terms in the form of power series to represent given function. We should calculate the function value f 0 , and some successive derivatives of the logarithmic function, to determine the n th order derivative. Properties of the power series expansion of the logarithmic function To graphically represent the power series or polynomial expansion of the logarithmic function we first calculate the coordinates of translations of the given sequence of polynomials. Since, The sequence of the polynomials and corresponding translations are written below and in the picture below shown are their graphs, where are also marked the x 0 , y 0 points that approach the origin as n tends to infinity. By plugging the coordinates of translations with changed signs into the polynomial expressed in general form, we get the source polynomial function.
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Finding the Sum of a Power Series

How to write functions as power series
How to write functions as power series
How to write functions as power series
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Power series intro (video) | Khan Academy

Since every polynomial function in the sequence, f 1 x , f 2 x , f 3 x ,. Now we apply the method and formulas that are revealed and explored under the ' Polynomial ' section. Thus, coordinates of translations,. Note that the above result proves the main property of the polynomial stating that, an n th degree polynomial function and all its successive derivatives to the n - 1 th order, have constant horizontal translation x 0. Thus, coordinates of translations, Note that the above result proves the main property of the polynomial stating that, an n th degree polynomial function and all its successive derivatives to the n - 1 th order, have constant horizontal translation x 0. Recall that an n th degree polynomial function and all its successive derivatives to the n - 1 th order, have constant horizontal translation x 0. Therefore as consequence, each x -intercept of odd polynomial in the sequence determines the abscissa of the only extreme minimum of succeeding even polynomial and the abscissa of the only point of inflection of succeeding odd polynomial, as shows the picture above.
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1.2: Properties of Power Series

A power series is a type of series with terms involving a variable. More specifically, if the variable is x , then all the terms of the series involve powers of x. As a result, a power series can be thought of as an infinite polynomial. Power series are used to represent common functions and also to define new functions.
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We are now going to look at some examples of determining power series representations for rational functions. Before we do so, we must recall a very important power series representation that we've already looked at, namely:. We will now demonstrate some examples of this useful technique below.
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