State and prove Cauchy theorem on limit (second theorem)

1.State and prove Cauchy theorem on limit (second theorem)

 Sol.

Statement: if <Sn> is a sequence s.t. Sn>0 for all n and lim Sn= l , 

Then lim (S1 ,S2 ......Sn)¹/n = l

Proof. Let us define a sequence < tn > s.t.

tn = log Sn for all n

Since lim Sn = l(L) therefore lim tn = log l

Lim ( t1 +t2 + ......+tn/ n) = log l

Lim (log S1 + log S2 + ......+log Sn/ n) = logl

Lim log (S1,S2 ,...... Sn) ¹/n = log l

Lim(S1,S2 ,...... Sn) ¹/n = l proved.

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