Introduction number power series:

The power series is one of the important topics in mathematics. When the terms of a sequence are connected by the sign +, it is called a series. Thus a1 + a2 + a3 + … an + … is an infinite series. The symbol Σan is used to denote a series.

 

Power series:

 

In mathematics, a power series (in individual variable) is an infinite series of the form

                  ∞
        f(x) = ∑ an (x-c)n= a0 + a1(x-c)1 +a2(x-c)2 +a3(x-c)3 + …….
               n=0

 Where an denoted the coefficient of the n th period, c is a constant, and x varies around c.

When c = 0,

            f(x) = ∑ anxn = a0 + a1x + a2x2 + a3x3 +…..
          

For example:

          Any polynomial preserve exist simply expressed as a power series around every center c, although one by means of most coefficients equivalent to zero. For example, the polynomial f(x) = x3 + 7x2 + 5x + 6 can be written as a power series around the center c = 0 as

          f(x)= 6 + 5x + 5x2 + x3 + 0x4 + 0x5 + ……

 

Examples of number power series:

 

Let us see some examples of number power series.

Example 1:

     Find a power series representation for 1/ x+3

Solution:

     `1/(x+3)`   = `1/ (3(1+x/3))` = `1/ (1- (-x/3))`

                      ∞
               = `1/3``(-x/3)^n`
                     n=0

                       ∞
               =1/3 ∑ `((-1)^n x^n)/ 3^n`
                    n=0

                  ∞
               = ∑  `1/3` ` ((-1)^n x^n)/3^n`
                n=0

                ∞
             = ∑   `((-1)^n x^n)/(3^n+1)`       which converges |-x/3| < 1
              n=0

         i.e., for |x| < 3

Example 2:

       Find a power series representation of `x^2/ (x+3)`

Solution:

       x2/x+3 = x2 (1/ x+3)

                   
             = x2`((-1)^n x^n)/ (3^(n+1))`
                   n=0

               
             = ∑ `((-1)^n )/ (3^(n+1)) . (x^n+2)`
              n=0

Which converges for |x| < 3.

Example 3:

       Find a power series representation for f(x) = ln (6 –x) and determine the radius of convergence

Solution:

       ln ( 6-x) = ln [ 6 (1- x/6)]

                   = ln 6 + ln ( 1- x/6)

                                 ∞
                  = ln 6 +  - ∑ `(x/6)^n /n`
                                n=0

                             ∞
                 = ln 6 - ∑ `x^n / (n6^n)`
                          n=0

Which converges for -1 ≤ x/6 ≤ 1

                         `rArr`   -6 ≤ x ≤ 6

Thus, the radius of convergence is 6.

These examples of number power series.