Series Convergent Calculator

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An infinite sequence (an) is called convergent if limit n tends to infinity an exists and is finite.
If this limit is not finite the sequence is called Divergent.
 
A series convergence calculator is used to find out the sum of the sequence and for determining convergence and divergence among series.
 
Illustration 1.
 
The sequence 1, 1/2,1/3……..1/n is convergent since an = 1/n → 0 as n tends to 0 (Zero)
 
Illustration 2.
 
The sequence 1, 2, 3, 4……n is divergent since an = n   →    Infinity.

Example 1.
Find the limit of sequence whose nth term is (-1n/n)
 
an= nth term = (-1n/n) = +- 1/n →0 as n tends to ∞


 
Example 2. Calculate Lt      an = ½+1+3/2+….+n/2.
l→∞n2/4+n+3
 
Solution
 
  Lt            ½+1+3/2+….+n/2
n→∞      n2/4+n+3
 
Since in the problem its given Limit n→∞anwe are just putting the value of an from the problem.
We will take ½ common from the numerator and ¼ common from the denominator.
Now we get,
 
 Lt    ½ (1+2+3+…..+n)
n→∞       1/4 (n2+4n+12)
We know that  1+2+3+…n is equal to n(n+1) hence we put this value in the numerator and take n2
                                                                        2
common in the denominator so we get.
 
Lt      ½*n (n+1)
n→∞   ___               2_________               
n2/4(1+4/n+12/n2)
Now we take the common factors and cancel them.
Lt      1*(1+1/n)
n→∞    1+4/n+12/n2
 
Since limit n →∞ we get,
    1*(1+0)____ = 1
  1+4*0+12*0
Hence limit n→∞, anequals 1.

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