6.2 Properties of power series  (Page 21/10)

In the preceding section on power series and functions we showed how to represent certain functions using power series. In this section we discuss how power series can be combined, differentiated, or integrated to create new power series. This capability is particularly useful for a couple of reasons. First, it allows us to find power series representations for certain elementary functions, by writing those functions in terms of functions with known power series. For example, given the power series representation for $f\left(x\right)=\frac{1}{1-x},$ we can find a power series representation for $f^{\prime }\left(x\right)=\frac{1}{{\left(1-x\right)}^2}.$ Second, being able to create power series allows us to define new functions that cannot be written in terms of elementary functions. This capability is particularly useful for solving differential equations for which there is no solution in terms of elementary functions.

Combining power series

If we have two power series with the same interval of convergence, we can add or subtract the two series to create a new power series, also with the same interval of convergence. Similarly, we can multiply a power series by a power of x or evaluate a power series at $x^m$ for a positive integer m to create a new power series. Being able to do this allows us to find power series representations for certain functions by using power series representations of other functions. For example, since we know the power series representation for $f\left(x\right)=\frac{1}{1-x},$ we can find power series representations for related functions, such as

In [link] we state results regarding addition or subtraction of power series, composition of a power series, and multiplication of a power series by a power of the variable. For simplicity, we state the theorem for power series centered at $x=0.$ Similar results hold for power series centered at $x=a.$

Proof

We prove i. in the case of the series ${\displaystyle \sum _{n=0}^{\infty }\left(c_n{x}^n+d_n{x}^n\right)}.$ Suppose that ${\displaystyle \sum _{n=0}^{\infty }{c}_n{x}^n}$ and ${\displaystyle \sum _{n=0}^{\infty }{d}_n{x}^n}$ converge to the functions f and g , respectively, on the interval I . Let x be a point in I and let $S_N\left(x\right)$ and $T_N\left(x\right)$ denote the N th partial sums of the series ${\displaystyle \sum _{n=0}^{\infty }{c}_n{x}^n}$ and ${\displaystyle \sum _{n=0}^{\infty }{d}_n{x}^n},$ respectively. Then the sequence $\left\{S_N\left(x\right)\right\}$ converges to $f\left(x\right)$ and the sequence $\left\{T_N\left(x\right)\right\}$ converges to $g\left(x\right).$ Furthermore, the N th partial sum of ${\displaystyle \sum _{n=0}^{\infty }\left(c_n{x}^n+d_n{x}^n\right)}$ is

Because

we conclude that the series ${\displaystyle \sum _{n=0}^{\infty }\left(c_n{x}^n+d_n{x}^n\right)}$ converges to $f\left(x\right)+g\left(x\right).$