Power Series

A power series is a special type of infinite series representing a mathematical function in the form of an infinite series that either converges or diverges. Whenever there is a discussion of power series, the central fact we are concerned with is the convergence of a power series. The convergence of a power series depends upon the variable of the power series.

The power series of a single variable converges within the radius of convergence, which means that within the extent of this radius or region of convergence, all the variable values less than the radius tend to converge to a point.

Table of Contents:

Definition of a Power Series

A series (centered at 0) of the form

\(\begin{array}{l}sum_{n=0}^{\infty}a_{n}x^{n}=a_{0}+a_{1}x+a_{2}x^{2}+…+a_{n}x^{n}\end{array} \)

where x is a continuous variable, and an is coefficient of x (independent of x) is called a power series.

Another definition of power series is, let {an} be a sequence of real or complex numbers and c be any real number, then the power series centered at c is given by

\(\begin{array}{l}\sum_{n=0}^{\infty}a_{n}(x-c)^{n}=a_{0}+a_{1}(x-c)+a_{2}(x-c)^{2}+…+a_{n}(x-c)^{n}\end{array} \)

Some important facts about power series:

  • Every power series is convergent for x = 0 irrespective of the value of the coefficient.
  • A power series may be
    • Nowhere convergent – if the power series is not convergent for any value of x other than x = 0.
    • Everywhere convergent – if for all values of x, the power series is convergent.
    • Convergent for some values, whereas divergent for others.
  • The set of all points x for which the power series is convergent is called the region of convergence.

Examples of Power Series

A polynomial function can be easily represented as power series, let f(x) = x3 -2x2 + 3x + 5, then f(x) can be represented as a power series as

f(x) = 5 + 3x + (-2)x2 + 1.x3 + 0x4 + ….+ 0xn

Where f(x) converges to zero x is equal to the roots of the given cubic polynomial.

The trigonometric and exponential functions are expressed in the form of power series.

sin x = x – x3/3! + x5/5! – x7/7! + …

ex = 1 + x + x2/2! + x3/3! + x4/4! + …

Logarithmic functions are also expressed as power series.

Radius of Convergence of a Power Series

For any real 0 ≤ R ≤ ∞, if the power series

\(\begin{array}{l}\sum_{n=0}^{\infty}a_{n}(x)^{n}\end{array} \)
converges for |x | < R and diverges for |x| > R, then the number R is called the radius of convergence for the given power series. The interval (-R, R) is called the interval of convergence for the power series.

The radius of convergence of a power series is given by

\(\begin{array}{l}R = \displaystyle \lim_{n \to \infty }\left| \frac{a_{n}}{a_{n+1}}\right|\end{array} \)

The radius of convergence is also given by

\(\begin{array}{l}R^{-1} = \displaystyle \lim_{n \to \infty }sup\sqrt[n]{\left|a_{n}\right|}\end{array} \)

The above is known as Hadamard Theorem.

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Properties of Power series

  • If the two power series
    \(\begin{array}{l}f(x) = \sum_{n=0}^{\infty}a_{n}(x-c)^{n}\end{array} \)
    and
    \(\begin{array}{l}g(x) = \sum_{n=0}^{\infty}b_{n}(x-c)^{n}\end{array} \)
    have same interval of convergence (-R, R) then
    \(\begin{array}{l}f(x) \pm g(x)=\sum_{n=0}^{\infty}(a_{n}\pm b_{n})x^{n}\end{array} \)
  • If the two power series
    \(\begin{array}{l}f(x) = \sum_{n=0}^{\infty}a_{n}(x-c)^{n}\end{array} \)
    and
    \(\begin{array}{l}g(x) = \sum_{n=0}^{\infty}b_{n}(x-c)^{n}\end{array} \)
    have same interval of convergence (-R, R) then
    \(\begin{array}{l}f(x) g(x)=f(x)g(x)=\sum_{n=0}^{\infty}c_{n}x^{n}\end{array} \)
    where
    \(\begin{array}{l}c_{n}=\sum_{k=0}^{\infty}a_{k}b_{n-k}\end{array} \)
  • A power series is a continuous function of x within its interval of convergence.
  • A power series can be integrated term by term within the limits of (-R, R).
  • Uniqueness of power series: If two power series have same radius of convergence, and converges to the same function then the power series are identical.

Solved Examples on Power Series

Example 1:

Find the radius of convergence of the series x + x2/22 + (2!/33)x3 + (3!/44)x4 + …..

Solution:

For the given power series x + x2/22 + (2!/33)x3 + (3!/44)x4 + …..

The coefficient an = (n – 1)!/nn

The radius of convergence

\(\begin{array}{l}R = \displaystyle \lim_{n \to \infty }\left| \frac{a_{n}}{a_{n+1}}\right|\end{array} \)

\(\begin{array}{l}= \displaystyle \lim_{n \to \infty }\left| \frac{(n-1)!}{n^{n}}\frac{(n+1)^{n+1}}{n!}\right| \end{array} \)

\(\begin{array}{l}= \displaystyle \lim_{n \to \infty }\left ( 1+\frac{1}{n} \right )^{n+1} \end{array} \)
= e

So, the radius of convergence is e.

Example 2:

Find the radius of convergence for the power series

\(\begin{array}{l}\sum_{n=0}^{\infty} \frac{2^{n}}{n}(4x-8)^{n}\end{array} \)

Solution:

For the given power series

\(\begin{array}{l}\sum_{n=0}^{\infty} \frac{2^{n}}{n}(4x-8)^{n}\end{array} \)

The coefficient an = 2n/n

The radius of convergence

\(\begin{array}{l}R = \displaystyle \lim_{n \to \infty }\left| \frac{a_{n}}{a_{n+1}}\right|\end{array} \)

\(\begin{array}{l}= \displaystyle \lim_{n \to \infty }\left| \frac{2^{n}}{n}\frac{n+1}{2^{n+1}}\right| \end{array} \)

\(\begin{array}{l}=\frac{1}{2}\displaystyle \lim_{n \to \infty }\left ( 1+\frac{1}{n} \right )= \frac{1}{2}\end{array} \)

Therefore, |4x – 8| < ½

Or, -½ < 4(x – 2) < ½

Or, -⅛ < (x – 2) < ⅛

Thus, the radius of convergence is 1/8.

Frequently Asked Questions on Power Series

Q1

What is a Power series?

A power series is an infinite series of increasing power of a variable used to express different mathematical functions.

Q2

What is the radius of convergence of a power series?

The radius of convergence of a power series is a real number 0 ≤ R ≤ ∞, such that if |x| R then the power series diverges.

Q3

What is meant by the region of convergence for a power series?

The region of convergence of a power series is the set of all the values of x of the power series for which the series converges.

Q4

What is the example of a power series?

Any trigonometric, logarithmic, exponential, geometric series and polynomial function can be expressed as a power series. For example, sin x = x – x3/3! + x5/5! – x7/7! + …

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