Difference Between Power Series and Taylor Series

Power Series vs Taylor Series
 

In mathematics, a real sequence is an ordered list of real numbers. Formally, it is a function from the set of natural numbers in to the set of real numbers. If a_nis the n^th term of a sequence, we denote the sequence by or by a₁, a₂, …,a_n, … .For example, consider the sequence 1, ½, ⅓, …, ¹/_n, … . It can be denoted as {1/n}.

It is possible to define a series using sequences. A series is the sum of the terms of a sequence. Therefore, for each sequence, there is an associated sequence and vice-versa. If {a_n} is the sequence under consideration, then, the series formed by that sequence can be represented as:                                           

 Thus, in the above example, the associated series is 1+¹/₂+¹/₃+ … + ¹/_n + … . 

As the names suggest, the power series is a special type of series and it is extensively used in Numerical Analysis and related mathematical modelling. Taylor series is a special power series that provides an alternative and easy-to-manipulate way of representing well-known functions.

What is Power series?

A power series is a series of the form

which is convergent (possibly) for some interval centered at c. The coefficients a_n can be real or complex numbers, and is independent of x; i.e. the dummy variable.

For example, by setting a_n= 1 for each n, and c = 0, the power series 1+x+x²+…..+ xⁿ+… is obtained. It is easy to observe that when x ε (-1,1), this power series converges to  1/(1-x).          

A power series converges when x = c. The other values of x for which the power series converges will always take the form of an open interval centred at c. That is, there will be a value 0≤ R ≤ ∞ such that for each x satisfying |x-c|≤R, the power series is convergent and for each x  satisfying |x-c|>R, the power series is divergent. This value R is called radius of convergence of the power series (R can take any real value or positive infinity).

Power series can be added, subtracted, multiplied and divided using the following rules. Consider the two power series:       

                               .

Then,                 

i.e. like terms are added or subtracted together. Also, it is possible to multiply and divide the two power series using the identity,

What is Taylor series?

Taylor series is defined for a function f(x) that is infinitely differentiable on an interval. Assume f(x) is differentiable on an interval centred at c. Then the power series which is given by

is called the Taylor series expansion of the function f(x) about c. (Here f⁽ⁿ⁾(c) denote the n^th derivative at x = c). In Numerical Analysis, a finite number of terms in this infinite expansion are used in calculating values at points where the series is convergent to the original function.

A function f(x) is said to be analytic in the interval (a, b), if for each x ε (a,b), the Taylor series of f(x) converges to the function f(x). For example, 1/(1-x) is analytic on (-1,1), since its Taylor expansion 1+x+x²+…..+ xⁿ+…  converges to the function on that interval, and e^x  is analytic everywhere, since the Taylor series of e^xconverges to e^x for each real number x.

What is the difference between Power series and Taylor series?

1. Taylor series is a special class of power series defined only for functions which are infinitely differentiable on some open interval.

2. Taylor series take the special form

whereas, a power series can be any series of the form