Question

State and prove cauchys first theorem on limits

Answer 1

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)


show that limn→∞(1 + 1/22+1/32+…..+1/n2)/n  = 0


Comparing to eq.(1),


Xn=(1 + 1/22+1/32+…..+1/n2)/n         


Here an = 1/n2


limn→∞(1/n2) = 0

hence 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
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