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The Editors of Encyclopaedia Britannica
power series, in mathematics, an infinite series that can be thought of as a polynomial with an infinite number of terms, such as 1 + x + x2 + x3 +⋯. Usually, a given power series will converge (that is, approach a finite sum) for all values of x within a certain interval around zero—in particular, whenever the absolute value of x is less than some positive number r, known as the radius of convergence. Outside of this interval the series diverges (is infinite), while the series may converge or diverge when x = ± r. The radius of convergence can often be determined by a version of the ratio test for power series: given a general power series a0 + a1x + a2x2 +⋯, in which the coefficients are known, the radius of convergence is equal to the limit of the ratio of successive coefficients. Symbolically, the series will converge for all values of x such that 
For instance, the infinite series 1 + x + x2 + x3 +⋯ has a radius of convergence of 1 (all the coefficients are 1)—that is, it converges for all −1 < x < 1—and within that interval the infinite series is equal to 1/(1 − x). Applying the ratio test to the series 1 + x/1! + x2/2! + x3/3! +⋯ (in which the factorial notation n! means the product of the counting numbers from 1 to n) gives a radius of convergence of so that the series converges for any value of x.
Britannica Quiz
Numbers and Mathematics
Most functions can be represented by a power series in some interval (see Click Here to see full-size tabletable). Although a series may converge for all values of x, the convergence may be so slow for some values that using it to approximate a function will require calculating too many terms to make it useful. Instead of powers of x, sometimes a much faster convergence occurs for powers of (x − c), where c is some value near the desired value of x. Power series have also been used for calculating constants such as π and the natural logarithm base e and for solving differential equations.
FAQs
Polynomials are simply finite power series. That is, a polynomial is a power series where the ak are zero for all k large enough. We can always expand a polynomial as a power series about any point x0 by writing the polynomial as a polynomial in (x−x0).
What is the general form of the power series? ›
Power series is a sum of terms of the general form aₙ(x-a)ⁿ.
What is convergence of power series functions? ›
Convergence of a Power Series. Since the terms in a power series involve a variable x, the series may converge for certain values of x and diverge for other values of x. For a power series centered at x=a, the value of the series at x=a is given by c0. Therefore, a power series always converges at its center.
What is the condition for power series to converge? ›
If R = 0, then the series converges only for x = 0. If R > 0, then the series converges absolutely for every x ∈ R with |x| < R, because it converges for some x0 ∈ R with |x| < |x0| < R. Moreover, the definition of R implies that the series diverges for every x ∈ R with |x| > R.
Do power series always converge? ›
Therefore, a power series always converges at its center. Some power series converge only at that value of x. Most power series, however, converge for more than one value of x. In that case, the power series either converges for all real numbers x or converges for all x in a finite interval.
Why is power series not a polynomial? ›
The formal polynomial expression is a function because you can always substitute a numerical value for x and get a number. That's not necessarily the case for a power series, since there may well be values for x for which the infinite series does not converge.
What is the power of a polynomial example? ›
Classification Based on Degree of Polynomial
| Polynomials | Degree | Examples |
|---|
| Linear Polynomial | Polynomials with Degree 1 | x + 8 |
| Quadratic Polynomial | Polynomials with Degree 2 | 3x2 - 4x + 7 |
| Cubic Polynomial | Polynomials with Degree 3 | 2x3 + 3x2 + 4x + 6 |
| Quartic Polynomial | Polynomials with Degree 4 | x4-16 |
2 more rowsPolynomials are one of the simplest functions to differentiate. When taking derivatives of polynomials, we primarily make use of the power rule. d d x f ( x ) = n x n − 1 .
What are the three cases of power series? ›
The Radius and Interval of Convergence of a Power Series
| Case | Radius ofConvergence |
|---|
| 1. Series converges only at x = x 0 . | R c = 0 |
| 2. Series absolutely converges for all . | R c = ∞ |
| 3. Series absolutely converges if | x − x 0 | < R c , and diverges if | x − x 0 | > R c . | R c > 0 |
Nov 29, 2023
Power series can be used to define functions and they allow us to write functions that cannot be expressed any other way than as “infinite polynomials.” An infinite series can also be truncated, resulting in a finite polynomial that we can use to approximate functional values.
If a series is a p-series, with terms 1np, we know it converges if p>1 and diverges otherwise. If a series is a geometric series, with terms arn, we know it converges if |r|<1 and diverges otherwise.
What is an example of a power series? ›
A power series looks like a polynomial with infinitely many terms, so one example of a power series might be 1 + x + x^2 + x^3...
How to prove convergence of power series? ›
The way to determine convergence at these points is to simply plug them into the original power series and see if the series converges or diverges using any test necessary. This series is divergent by the Divergence Test since limn→∞n=∞≠0 lim n → ∞ n = ∞ ≠ 0 .
What order to watch power series? ›
The chronological order to watch the shows is: Power Book III: Raising Kanan, all seasons of Power, Power Book II: Ghost season 1, Power Book IV: Force season 1, Power Book II: Ghost season 2, Power Book IV: Force season 2, Power Book II: Ghost season 3.
How do you tell if a series will converge? ›
If a series is a p-series, with terms 1np, we know it converges if p>1 and diverges otherwise. If a series is a geometric series, with terms arn, we know it converges if |r|<1 and diverges otherwise. In addition, if it converges and the series starts with n=0 we know its value is a1−r.
Where does a power series diverge and converge? ›
assuming that lim|an+1|/|an| exists. Then the series converges if L|x|<1, that is, if |x|<1/L, and diverges if |x|>1/L.
Where does a power series converge conditionally? ›
convergence. The power series converges absolutely for any x in that interval. Then we will have to test the endpoints of the interval to see if the power series might converge there too. If the series converges at an endpoint, we can say that it converges conditionally at that point.
How do you prove a power series converges uniformly? ›
Use the Weierstrass M-test
The Weierstrass M-test is a powerful tool to prove uniform convergence. It essentially states that if we can find a sequence of positive real numbers such that | a n ( z − a ) n | ≤ M n for all z ∈ K and ∑ n = 0 ∞ M n converges, then the power series converges uniformly on .
so that the series converges for any value of x.
table). Although a series may converge for all values of x, the convergence may be so slow for some values that using it to approximate a function will require calculating too many terms to make it useful. Instead of powers of x, sometimes a much faster convergence occurs for powers of (x − c), where c is some value near the desired value of x. Power series have also been used for calculating constants such as π and the natural logarithm base e and for solving differential equations.