L'Hospital's Rule

L'Hospital's Rule is a method for finding the value of a function using derivatives. This method is use when, the value at a point don't exist.

Suppose there are continuous function f(x) and g(x) that are both zero at x=a. Then, the limit   cannot be found by substituting x=a since it give 0/0 which cannot be evaluated we use 0/0 as a notation for an expression known as an indeterminate form may be evaluated by cancellation or rearrangement of terms. However, this does not always work.
      For example, how would you evaluate   ? Obviously inserting x=0 will give an indeterminate form of 0/0 and, in this case you can neither use algebraic manipulation nor rearrangement of terms to reduce this expression into a form that yield a valid limit.

L'Hospital's Rule

Statement: If f and g are differentiable function such that

1.   or

     ;

2. f'(x), g'(x) both exist and
g'(x)≠0 ∀ x∈(a-δ,a+δ), δ>0 except possibly at x=a;

3.  exist
Then

Examples

Q1. Evaluate the limit : .
Ans.
       Directly applying x=0 leads the limit to an indeterminate form. Since both the term in the numerator and the term in the denominator are zero at x=0 and since sin x and x are both differentiable at x=0, we can use L'Hospital's Rule.

          .

Q2. Evaluate 
Ans.
                              

Directly applying x=0 leads the limit to an indeterminate form. Since both the term in the numerator and the term in the denominator are zero at x=0 and since both are differentiable at x=0, we can use L'Hospital's Rule.

    =  

    =  

                          
    = 1.

Q3. Evaluate  .
Ans.
                                

Directly applying x=0 leads the limit to an indeterminate form. Since both the term in the numerator and the term in the denominator are zero at x=0 and since both are differentiable at x=0, we can use L'Hospital's Rule.
       =                      
     
Since after applying L'Hospital's Rule we are getting 0/0 form. So we apply the rule again till we didn't get 0/0 form.

       =                       

       =                       

       =                      

       =    
       =  120×0 
       =     0

Divergent sequence and series

In Mathematics, a divergent sequence is an infinite sequence that is not convergent, meaning that we never find a greatest number in a sequence from which the whole sequence terms are smaller.

Divergent sequence

1. A sequence {aₙ} is said to diverge to ∞ if given k>0, however large, there exist a positive integer m (depending upon k) such that
              
we write it as
              

As shown in the figure we have taken aₙ on y-axis and its subscript on x-axis. Let suppose that k is a very large number. If the sequence need to diverge, then we will always find a term in a sequence which is greater then k. Therefore the sequence diverge to +∞.

2. A sequence {aₙ} is said to diverge to -∞ if given k>0, however large, there exist a positive integer m (depending upon k) such that
             
we write it as

            

As shown in the figure we have taken aₙ on y-axis and its subscript on x-axis. Let suppose that k is a very large number. If the sequence need to diverge, then we will always find a term in a sequence which is smallest  then -k. Therefore the sequence diverge to -∞.

Divergent series

The series ∑aₙ is said to diverge to ∞ if the sequence {sₙ} diverge to ∞. We write 

                           .

The series ∑aₙ is said to diverge to -∞ if the sequence {sₙ} diverge to -∞. We write

                           .

Infinite series

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

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

partial sum

        we define
                s₁=a₁
                s₂=a₁+a₂
                s₃=a₁+a₂+a₃
                ...  ...   ...   ...  ....
                ...  ...   ...   ...  ....
                sₙ=a₁+a₂+a₃+....+aₙ= 

    There s's are called the partial Sums of the series ∑aₙ. The nth partial sum is denoted by Sₙ or sₙ.

Behaviour of an Infinite series

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

        1. Convergent series : The series ∑aₙ is said to converge if the sequence {sₙ} of its partial sums converges to s and s is called the sum of the convergent infinite series 

          . we write  
         here sₙ→s as n→∞

        2. Divergent series : The series ∑aₙ is said to diverge to ∞ if sequence {sₙ} diverges to ∞. We write

                               
       The series ∑aₙ is said to diverge to -∞ if sequence {sₙ} diverge to -∞. We write 

                               

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

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

       5. Conditionally convergent series : The series ∑aₙ is said to conditionally convergent if ∑aₙ is convergent but  ∑|aₙ| 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 aₙ 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 {aₙ} is monotone decreasing sequence of positive terms and converges to zero, then   is convergent

Three points you need to know

•   aₙ>0   ∀  n
•   aₙ ≥ aₙ₊₁  ∀  n and
•   aₙ→0 as n→∞  

3. Comparison test for series of positive terms

    Test 1. Let ∑aₙ and ∑bₙ  be positive terms series. Let k be a positive constant independent of n and m be a fixed positive integer.

•    If aₙ ≤ kbₙ  ∀  n ≥ m and ∑bₙ is convergent, then ∑aₙ is convergent.
•    If aₙ ≥ kbₙ ∀  n ≥ m and ∑bₙ is divergent, then ∑aₙ is divergent.

    Test 2. Let ∑aₙ and ∑bₙ be two positive terms series.

     * If   .l being a finite non- zero constant, then ∑aₙ and ∑bₙ both converge or diverge together.

    * If  and∑bₙ converges, then ∑aₙ also converges.

    * If   and∑bₙ diverges, then ∑aₙ also diverges.

4. P-Test

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

5. Cauchy's Root Test

If , aₙ≥, the series  is convergent if l<1 and divergent if l>1.

6.D′Alembert′s Ratio Test

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

7. Raabe's Test

If ∑aₙ be a positive term series, then ∑aₙ converges if  and diverges if 

8. Gauss′s Test

If aₙ>0 and  , then ∑aₙ converges for μ>1 and diverges for μ≤1.

9. Logarithmic Test

If aₙ>0, then the series ∑aₙ

•    converges if   and
•    diverges if