(adsbygoogle = window.adsbygoogle || []).push({}); The problem statement, all variables and given/known data

Let a_{n}be a sequence of real numbers. For what values of x does lim a_{n}x^{n}exist?

The attempt at a solution

Let us suppose that lim a_{n}x^{n}exist and is equal to b. What can we say about x? Hmm...there is a monotonic subsequence that converges to b, say [itex]a_{k_n}x^{k_n}[/itex]. If this is an increasing sequence, we have that

[tex]

a_{k_n}x^{k_n} \le a_{k_{n+1}}x^{k_{n+1}}

[/tex]

or equivalently

[tex]

\frac{a_{k_n}}{a_{k_{n+1}}} \le x^{k_{n+1} - k_n}

[/tex]

Unfortunately I don't get an inequality in terms of x alone. How do I proceed from here? Perhaps I need a further assumption, like [itex]k_{n+1} - k_n = 1[/itex] for all n?

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# Homework Help: When does lim a_n x^n exist?

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