Power Series

Power series in mathematics is an infinite series of the form $\sum_{n=0}^{\infty}a_n (x - c)^n$. Where $a_n$ is the $n^{th}$ term coefficient and $c$ is a constant.

$\sum_{n=0}^{\infty}a_n(x - c)^n = a_0 + a_1(x - c)^1 + a_2(x - c)^2 + a_3(x - c)^3 + .............$

Any polynomial can be expressed as a power series around any $c$, but all but a small number of the coefficients will be zero since a power series by definition has infinitely many terms. When the absolute value of $x$ is less than some positive number $r$, known as the radius of convergence, a given power series will usually converge, that is, the power series will reach a finite sum for all the given values of $x$ within a certain interval around zero in particular. Outside of this interval, the series diverges, reaching an infinite number, while when $x = \pm r$, the series can converge or diverge. A variant of the ratio test for power series may also be used to evaluate the radius of convergence.

Power Series Examples

Every polynomial can be expressed as a power series around $c$, but all but a small number of the coefficients will be zero because, by definition, a power series has infinite terms. 

For example the polynomial $f(x) = x^3 + 7x^2 + 4x + 3$ can be written in a power series around the $c = 0$ as follows:

$f(x) = 3 + 4x + 7x^2 + 1x^3 + 0x^4 + 0x^5 + .........$

The power series formula for a geometric series formula will be as follows: 

$\dfrac{1}{1 - x} = \sum_{n=0}^{\infty}x^n = 1 + x + x^2 + x^3 + x^4 + .................. $

Power series in mathematics for an exponential function is 

$e^x = \sum_{n=0}^{\infty} \dfrac{x^n}{n!} = 1 + \dfrac{x}{1} + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \dfrac{x^4}{4!} + ................$

Power series in mathematics for sine functions is

$\sin(x) = \sum_{n=0}^{\infty} \dfrac{(-1)^n x^{2n+1}}{(2n+1)!}= x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dfrac{x^7}{7!} + \dfrac{x^9}{9!} -.............$

Power series in mathematics for cosine function is 

$\cos(x) = \sum_{n=0}^{\infty} \dfrac{(-1)^n x^{2n}}{2n!} = 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \dfrac{x^6}{6!} + \dfrac{x^8}{8!} - .............$

The power series in mathematics of a logarithmic function is 

$ln(1+x) = \sum_{n=0}^{\infty} \dfrac{(-1)^{n-1} x^n}{n} = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dfrac{x^4}{4} + \dfrac{x^5}{5} - .............$

The power series in mathematics of an inverse tangent function is 

$tan^{-1}(x) = \sum_{n=0}^{\infty} \dfrac{(-1)^n x^{2n+1}}{2n+1} = x - \dfrac{x^3}{3} + \dfrac{x^5}{5} - \dfrac{x^7}{7} + \dfrac{x^9}{9} - .............$

Radius of Convergence

For certain values of the variable $x$, such as $x = c$, a power series is convergent. For other $x$ values, the series can diverge. If $c$ isn't the only point of convergence, there's always a number $r$ with $0 < r \leq \infty$ such that the series converges when $|x – c| < r$ and diverges when $|x – c| > r$. The radius of convergence of the power series is denoted by the number $r$. 

The radius of convergence $r$ for the power series is given as follows,

$r = \lim_{n \to \infty} inf |an|^{-\frac{1}{n}}$ 

Or 

$r^{-1} = \lim_{n \to \infty} sup |a_n|^{\frac{1}{n}} $

Here $\lim\,inf$ and $\lim\,sup$ are limit inferior and limit superior which are limiting bounds on the sequence.

Operations on Power Series

We can perform basic operations on power series along with the complex calculus operations too. Here let us have a look at a few of the important operations performed on the power series.

Addition and Subtraction of the Power Series

When two functions f and g are decomposed into power series around the same centre c, termwise addition and subtraction can be used to obtain the power series of the sum or difference of the functions.

If $f(x) = \sum_{n=0}^{\infty} a_n(x-c)^n$ and $g(x) = \sum_{n=0}^{\infty} b_n(x-c)^n$ are two power series then the addition and subtraction are as follows:

$f(x) \pm g(x) = \sum_{n=0}^{\infty}(a_n \pm b_n)(x-c)^n$

Multiplication and Division of the Power Series

If $f(x) = \sum_{n=0}^{\infty}a_n(x - c)^n$ and $g(x) = \sum_{n=0}^{\infty}b_n(x - c)^n$ are two power series then the multiplication of the two power series is as follows:

$f(x) g(x) = \left(\sum_{n=0}^{\infty}a_n(x-c)^n \right) \left(\sum_{n=0}^{\infty} b_n(x-c)^n \right) \\ f(x) g(x) = \sum_{i=0}^{\infty} \sum_{j=0}^{\infty} a_i b_j (x-c)^{i+j} \\ f(x) g(x) = \sum_{n=0}^{\infty} \left(\sum_{i=0}^{\infty} a_i b_{n-i} \right) (x-c)^n$

The sequence $\sum_{i=0}^{\infty} a_ib_{n-i}$ is known as the convolution of the sequences $a_n$ and $b_n$.

If $f(x) = \sum_{n=0}^{\infty} a_n(x-c)^n$ and $g(x) = \sum_{n=0}^{\infty} b_n(x-c)^n$ are two power series then the division of the two power series is as follows:

$\dfrac{f(x)}{g(x)} = \dfrac{ \sum_{n=0}^{\infty} a_n(x-c)^n}{\sum_{n=0}^{\infty} b_n(x-c)^n} = \sum_{n=0}^{\infty} d_n(x-c)^n$

Differentiation and Integration of the Power Series

When a function $f(x)$ is expressed as a power series $\sum_{n=0}^{\infty} a_n(x-c)^n$ , it can be differentiated on the interior of the convergence domain. It's simple to differentiate and integrate by treating each term separately.

Differentiation of the power series $f(x)\sum_{n=0}^{\infty} a_n(x-c)^n$ is as follows:

$f^1(x)\sum_{n=1}^{\infty} a_n n(x-c)^n = \sum_{n=0}^{\infty} a_{n+1}(n+1)(x-c)^n$

Integration of the power series $f(x) = \sum_{n=0}^{\infty} a_n(x-c)^n$ is as follows:

$\int{f(x)\,dx} = \sum_{n=0}^{\infty} \dfrac{a_n(x-c)^{n+1}}{n+1} = \sum_{n=1}^{\infty} \dfrac{a_{n-1}(x-c)^n}{n} +k$

Applications of Power Series

• Power series can be found as the Taylor series of infinitely differentiable functions in mathematical analysis. Every power series, according to Borel's theorem, is the Taylor series of some smooth function.

• Power series appear as generating functions in combinatorics and as the Z-transform in electronic engineering, in addition to their role in mathematical analysis.

• An example of a power series is also the well-known decimal notation for real numbers.

• The definition of p-adic numbers is closely related to that of a power series in number theory.

Conclusion

Power series are useful tools that can be used to extend other functions, solve equations, test convergence intervals, and serve as trial functions in a variety of engineering applications. Taylor's Series, which are extremely significant in numerical approximations, use power series.