The power series representation is
We find the power series representation of a function.
1 Power Series
In this section we will use power series to represent familiar functions. A power series representation of a function is a convergent power series whose sum is equal to the given function. Our motivation will be the geometric power series that we saw in the last section, which converges when . Moreover, since this is a geometric series, we can find the sum of this series and this sum gives us the primary example of a power series representation of a function.
example 1 Find a power series representation for the function The sum of the
geometric series as long as . Thus, the power series representation of the function is
given by and this representation is valid as long as .
The result of this example will be used throughout the remainder of this section.
example 2 Find a power series representation for the function and determine the
interval on which this representation is valid.
The function in this example is twice the function in the previous example, so we will simply double the power series representation from that example. The power series representation is and this representation is valid as long as .
The function in this example is twice the function in the previous example, so we will simply double the power series representation from that example. The power series representation is and this representation is valid as long as .
example 3 Find a power series representation for the function and determine the
interval on which this representation is valid.
The function in this example is times the function in example 1, so we will simply multiply the power series representation from that example by . The power series representation is and this representation is valid as long as .
The function in this example is times the function in example 1, so we will simply multiply the power series representation from that example by . The power series representation is and this representation is valid as long as .
example 4 Find a power series representation for the function and determine the
interval on which this representation is valid.
Since , we can use the result of example 1 with in the place of . Thus, the power series representation is
Since , we can use the result of example 1 with in the place of . Thus, the power series representation is
This representation is valid as long as , which is equivalent to . Note the final form of the answer is the standard form of a power series,
example 5 Find a power series representation for the function and determine the
interval on which this representation is valid.
We can use the result of example 1 with in the place of . Thus the power series representation is This representation is valid as long as , which is equivalent to .
We can use the result of example 1 with in the place of . Thus the power series representation is This representation is valid as long as , which is equivalent to .
example 6 Find a power series representation for the function and determine the
interval on which this representation is valid.
The function in this example resembles the function in example 1 except that there is a ‘2’ in the denominator instead of a ‘1’. This is an important distinction and we need the ‘1’. To create it, we will factor ‘2’ from the denominator: Now we have times a function like the one in example 1 with in the place of . Thus, the power series representation is This representation is valid as long as which is equivalent to .
The function in this example resembles the function in example 1 except that there is a ‘2’ in the denominator instead of a ‘1’. This is an important distinction and we need the ‘1’. To create it, we will factor ‘2’ from the denominator: Now we have times a function like the one in example 1 with in the place of . Thus, the power series representation is This representation is valid as long as which is equivalent to .
In the next example, we combine the ideas of the previous examples.
example 7 Find a power series representation for the function and determine the
interval on which this representation is valid.
We begin by factoring a ‘4’ from the denominator: We can now apply the result of example 1 with in place of : This representation is valid as long as which is equivalent to which, in turn, is equivalent to .
We begin by factoring a ‘4’ from the denominator: We can now apply the result of example 1 with in place of : This representation is valid as long as which is equivalent to which, in turn, is equivalent to .
Here is a detailed, lecture style video on power series representation of functions:
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