Question

Cauchy first theorem on limit & eg

Answer 1

Cauchy’s theorems on limits
Cauchy’s first theorem on limits:
            If a sequence {an} converges to k, then the sequence {xn} also converges to k.
Where, xn=(a1+a2+a3+……+an)/n                         …………..(1)
Example: show that limn→∞(1 + 1/22 +1/32+…..+1/n2)/n  = 0
Sol. Comparing to eq.(1),
Xn=(1 + 1/22+1/32+…..+1/n2)/n         
Here an = 1/n2
limn→∞(1/n2) = 0hence according to Cauchy's first theorem,
This implies, {an=1/n2} converges to 0, and then so {xn} also converges to 0.
Hence,limn→∞(1+1/22+1/32+…..+1/n2)/n  = 0
 
Cauchy’s second theorem on limits:
If {an} is a sequence(should be positive terms) , then
            limn→∞(an)1/n = limn→∞(an+1/an)
Example: show that limn→∞(n)1/n = 1
Solution: here an=n
            an+1=n+1
So, an+1/an = (n+1)/n = 1+(1/n)
limn→∞(an+1/an)  = limn→∞{1+(1/n)} = 1+0 = 1so, according to Cauchy's second theorem,limn→∞(an)1/n = limn→∞(an+1/an) 
Hence , limn→∞(n)1/n = 1