Statement:
Let (an) be a sequence of positive terms lim n->infinity a power 1/n = lim n->infinity an+1/an. Provided the limit on the right hand side exist, whether finite (or) infinite.Proof:
Case(i):
lim n->infinity an+1/an = l (finite)
Let excellon > 0 be given, there exist m belongs to N. Such that l - 1/2 excellon < an+1/an < l+2 excellon for all n >= m.
Now, choose n >= m,
l - excellon/2 < am+1/am < l+excellon/2
l - excellon/2 < am+2/am < l+excellon/2
l - excellon/2 < an/an-1 < l+excellon/2
Multiplying these inequalities we get, (l - excellon/2)power n-m < an/am < (l+excellon/2)power n-m
am (l-excellon/2)power n/(l-excellon/2)power m < an (l+excellon/2)power n/(l+excellon/2)power m. am
k1 (l-excellon/2)power n < an < k2 (l+excellon/2)power n
Where k1,k2 are same constants.
Therefore, k1 power 1/n (l-excellon/2) < an power 1/n < k2 power 1/n (l+excellon/2)
Therefore, now, (k1 power 1/n (l-excellon/2)) -> l- excellon/2. [(k1)power 1/n -> l]
Therefore, there exist n1 belongs to N such that (l-excellon/2) - excellon/2 < k1 power 1/n (l-excellon/2) <
(l-excellon/2) + excellon/2 for all n >= n1.
Hence this Theorem.
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