Friday, 1 February 2019

Infinite series

An Infinite series is the sum of the values in an infinite sequence of number.

Geometric series

       Let {aâ‚™} be a sequence of real numbers.
       The expression a₁+a₂+a₃+a₄+.....+aâ‚™..... is called an infinite series and is denoted by  or by ∑aâ‚™ and aâ‚™ is called the nth term of the series.

partial sum

        we define
                s₁=a₁
                s₂=a₁+a₂
                s₃=a₁+a₂+a₃
                ...  ...   ...   ...  ....
                ...  ...   ...   ...  ....
                sâ‚™=a₁+a₂+a₃+....+aâ‚™= 

    There s's are called the partial Sums of the series ∑aâ‚™. The nth partial sum is denoted by Sâ‚™ or sâ‚™.

Behaviour of an Infinite series


        The behaviour of the infinity series ∑aâ‚™ is the same as that of the sequence {sâ‚™}. In other words, the series ∑aâ‚™ is said to be convergent, divergent or oscillating as the sequence {sâ‚™} converges, diverges or oscillates.

        1. Convergent series : The series ∑aâ‚™ is said to converge if the sequence {sâ‚™} of its partial sums converges to s and s is called the sum of the convergent infinite series 
          . we write  
         here sâ‚™→s as n→∞

        2. Divergent series : The series ∑aâ‚™ is said to diverge to ∞ if sequence {sâ‚™} diverges to ∞. We write
                               
       The series ∑aâ‚™ is said to diverge to -∞ if sequence {sâ‚™} diverge to -∞. We write 
                               

        3. Oscillatory series : The series ∑aâ‚™ is said to oscillate finitely or infinitely if the sequence {sâ‚™} oscillate finite ( or finitely ) if the sequence {sâ‚™} is bounded ( or unbounded ).

       4. Absolute convergent series : The series ∑aâ‚™ is said to converge absolutely if the series   i.e. the series |a₁|+|a₂|+|a₃|+.....+|aâ‚™|+.... is convergent.

       5. Conditionally convergent series : The series ∑aâ‚™ is said to conditionally convergent if ∑aâ‚™ is convergent but  ∑|aâ‚™| is not convergent.

       6. Non-convergent series : Series which diverge or oscillate are said to be non-convergent.

    Nth term test
If the series  is convergent, then 

Important. The converse of this theorem is false. The convergence of aâ‚™ to zero does not imply that the series  converges.

For example:- The harmonic series   diverges, although 
 Equivalently, if   or this limit does not exist, then the series   is divergent.

Test on infinite series


1. Infinite geometric series


        
  •    converges to  
  •    diverge to ∞ if r≥1
  •    oscillates finitely between 0 and 1 if r =-1
  •    oscillates infinitely if r<-1
  •    converges absolutely for | r |<1


2. Leibnitz test (alternating series)

 
  A series whose terms are alternatively positive and negative, is called an alternative series
  For examples   is an alternating series

Definition:- If {aâ‚™} is monotone decreasing sequence of positive terms and converges to zero, then   is convergent

Three points you need to know

  •   aâ‚™>0   ∀  n
  •   aâ‚™ ≥ aâ‚™₊₁  ∀  n and
  •   aâ‚™→0 as n→∞  

3. Comparison test for series of positive terms


    Test 1. Let ∑aâ‚™ and ∑bâ‚™  be positive terms series. Let k be a positive constant independent of n and m be a fixed positive integer.
  •    If aâ‚™ ≤ kbâ‚™  ∀  n ≥ m and ∑bâ‚™ is convergent, then ∑aâ‚™ is convergent.
  •    If aâ‚™ ≥ kbâ‚™ ∀  n ≥ m and ∑bâ‚™ is divergent, then ∑aâ‚™ is divergent.

    Test 2. Let ∑aâ‚™ and ∑bâ‚™ be two positive terms series.

     * If   .l being a finite non- zero constant, then ∑aâ‚™ and ∑bâ‚™ both converge or diverge together.

    * If  and∑bâ‚™ converges, then ∑aâ‚™ also converges.

    * If   and∑bâ‚™ diverges, then ∑aâ‚™ also diverges.

4. P-Test


The series 
converge if  p>1 and diverge if p≤1

5. Cauchy's Root Test


If , aâ‚™≥, the series  is convergent if l<1 and divergent if l>1.

6.D′Alembert′s Ratio Test


If , then the positive terms series ∑aâ‚™ converges if l<1 and diverges if l>1.

7. Raabe's Test


If ∑aâ‚™ be a positive term series, then ∑aâ‚™ converges if  and diverges if 

8. Gauss′s Test


If aâ‚™>0 and  , then ∑aâ‚™ converges for μ>1 and diverges for μ≤1.

9. Logarithmic Test


If aâ‚™>0, then the series ∑aâ‚™

  •    converges if   and
  •    diverges if  

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