Power Series And It's Theorems

Power Series And It's Theorems :

Definition Of Power Series :

                A series of the form

     ∞

     Σ aₙ(x-x₀)ⁿ = a₀+a₁(x-x₀)+a₂(x-x₀)²+......

    n=0

                                           + aₙ(x-x₀)ⁿ+....

is called a power series .

     where x is a continuous variable and the constants aₙ,x₀ are real and independent of x .

Remarks :

   --> If we change the variable x= t+x₀ ,              then the power series reduces to

                ∞

                Σ aₙ tⁿ  .

                n=0

   -->For x=0 , every power series is                      convergent , whatever the value of              co_efficients .

 --> A power series is either 

      (1) convergent for no value of x other              than x= 0 , then it is said to be                      nowhere convergent . 

      (2) convergent for all values of x and                is called everywhere convergent .

       (3) convergent for some values of x                   and diverges for others .

-->  The totality of point x for which a               power series converges is called                   Region Of Convergence .

Theorem _ 1 :

                                             ∞

            If a power series   Σ aₙ xⁿ converges

                                            n=0

for a particular value x₀ of x then it converges absolutely for all values of x for which |x| < |x₀|.

Proof :

              Let the series Σ aₙ x₀ⁿ converges .

Then nth term of the given series aₙx₀ⁿ must tends to 0 as n-->∞ . Therefore , we can find a number M>0 such that 

        |aₙx₀ⁿ|≤M for all n .

Then |aₙxⁿ|≤M|x/x₀|ⁿ .........(1)

Now , since |x|<|x₀| , then geometric series Σ|x/x₀|ⁿ converges . 

Hence 

from(1), the series Σ|aₙxⁿ| converges for all values of x for which |x| < |x₀| .

Theorem _2 :

                 The power series Σ aₙ xⁿ either 

(i) converges absolutely for all x .

or (ii) converges for x= 0 only .

or (iii) ∃ R > 0 such that the series                   converges absolutely , when |x| <  R and    diverges when |x|>R .

Proof : 

         Let x∈ + and P be the set of all x for which the given power series Σ aₙxⁿ converges . Then given series is definitely converges to 0 .

          ∴  0 ∈ P .

       Then P may or may not contain numbers other than 0 . If x₀ ∈ P then by previous theorem , every x with 0≤x≤x₀ is also in P . 

       If for all non_ negative real numbers are in P , then Σ aₙ xⁿ absolutely converges for all x . 

       On the other hand , if P does not contain all non_negative numbers , then it has a least upper bound R (R>0) . Let R>0 and |x₀|< R . 

      Then , to show that Σ aₙ x₀ⁿ absolutely converges .

    Let us choose R₀ such that |x₀|<R₀<R . Then R₀∈ P and so the series converges for x= R₀ . Then by previous theorem 

 Σ |aₙx₀ⁿ | converges .

    Finally , we prove if |x'|>R≥0 , the series can't converges for x= x' .

    Select R₀ with R<R₀<|x'| . If Σaₙx'ⁿ were to converge , then by previous theorem  Σ  aₙR₀ⁿ also converges , which contradicts the fact  R= P .