How to Solve and Apply Power Series in Mathematics
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.
FAQs on Power Series Explained: Concepts, Examples & Uses
1. What is a power series and what are its main components?
A power series is a type of infinite series that can be thought of as a polynomial with an infinite number of terms. The standard form is ∑n=0∞an(x - c)n. Its main components are:
- an: The coefficient of the n-th term.
- x: The variable.
- c: The constant 'center' of the series, around which it is expanded.
Essentially, it's a way to represent functions as an infinite sum of powers of (x-c).
2. What do the 'radius of convergence' and 'interval of convergence' of a power series signify?
A power series may not converge (yield a finite sum) for all values of x. The radius of convergence (r) defines a range around the center 'c' where the series is guaranteed to converge. The interval of convergence is the actual set of x-values for which the series converges. It is typically expressed as (c - r, c + r), but may also include one or both endpoints, which must be tested separately.
3. How can common functions like ex, sin(x), and cos(x) be represented as power series?
Many fundamental functions in mathematics can be expressed as a power series centered at c=0 (also known as a Maclaurin series). Key examples include:
- Exponential function (ex): ∑n=0∞ (xn/n!) = 1 + x + (x2/2!) + (x3/3!) + ...
- Sine function (sin(x)): ∑n=0∞ ((-1)n x2n+1)/(2n+1)! = x - (x3/3!) + (x5/5!) - ...
- Cosine function (cos(x)): ∑n=0∞ ((-1)n x2n)/(2n)! = 1 - (x2/2!) + (x4/4!) - ...
4. Why are power series so important in fields like engineering and physics?
Power series are a fundamental tool in applied mathematics for several reasons. Their importance lies in their ability to:
- Approximate complex functions: They allow engineers and physicists to approximate complicated functions with simpler polynomials, which are easier to compute and analyze.
- Solve differential equations: Many differential equations that model physical phenomena (like vibrations or heat flow) do not have simple solutions. Power series provide a method to find an approximate solution.
- Model physical behaviour: They are used in optics, quantum mechanics, and electrical engineering to model wave functions and field potentials.
5. What is the difference between a power series and a Taylor series?
This is a common point of confusion. The key difference is in how the coefficients are determined. A Taylor series is a specific type of power series where the coefficients (an) are precisely calculated from the derivatives of a given function at a center 'c'. A power series is the more general form, ∑an(x-c)n, where the coefficients can be any sequence of numbers. In short, every Taylor series is a power series, but not every power series is a Taylor series for some function.
6. How can an infinite number of terms in a power series add up to a single, finite value?
This is possible because of the mathematical concept of convergence. Within its interval of convergence, the magnitude of successive terms in a power series gets smaller and smaller, approaching zero very rapidly. While you are always adding more terms, the amount you add each time diminishes. The sum gets progressively closer to a specific finite number, known as the limit. Outside this interval, the terms do not shrink fast enough, and the sum grows infinitely (diverges).
7. Can you differentiate and integrate a power series?
Yes, one of the most powerful properties of a power series is that it can be differentiated and integrated term-by-term within its interval of convergence. This means you can find the derivative or integral of the function it represents by simply differentiating or integrating the polynomial-like expression term by term. The resulting series will have the same radius of convergence as the original series.
