Limit of infinite sequence raised to a real number power

In summary, the problem involves proving that the sequence {(an)x} converges to Lx, given that the sequence {an} converges to L. The key concepts to consider are the existence of an increasing rational sequence with limit x, and the fact that a monotone sequence is convergent if and only if it is bounded. Using the power function, it can be shown that the sequence {(an)x} is also monotone, and the least upper bound/greatest lower bound of this sequence can be found, leading to the conclusion that the entire sequence converges to Lx.
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Homework Statement



Let {an} be a sequence with positive terms such that lim an = L > 0.

Prove lim (an)x = Lx.

Homework Equations



If x is a real number, there exists an increasing rational sequence {rn} with limit x.

A monotone sequence {an} is convergent if and only if {an} is bounded.

ax = lim arn.

The Attempt at a Solution



I know that each of the individual elements of the sequence {(an)x} converge to (an)x, but I do not know what else I can do from here.

Any help would be greatly appreciated. Thanks!
 
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  • #2
It may help to know that every convergent sequence in [itex] \mathBB{R} [/itex] contains a monotone subsequence. The power function [tex] f(x) = x^a [/tex] preserves monotonicity (although it may reverse it for negative powers). What would be a bound for such a sequence? Why would it have to be the least upper bound/greatest lower bound? What does that say about the rest of your sequence?
 
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1. What does it mean to raise a limit of infinite sequence to a real number power?

When we say "limit of infinite sequence raised to a real number power," we are referring to the process of taking a sequence of numbers that continues infinitely and raising each number in the sequence to a specific real number. This results in a new sequence of numbers that may approach a finite limit or may continue to infinity depending on the original sequence and the real number power.

2. How is the limit of infinite sequence raised to a real number power calculated?

The calculation of the limit of infinite sequence raised to a real number power involves taking the limit of the original sequence and then raising it to the given real number. This can be done algebraically or by using mathematical techniques such as L'Hopital's Rule or the Squeeze Theorem.

3. What is the significance of raising a limit of infinite sequence to a real number power?

Raising a limit of infinite sequence to a real number power can help us understand the behavior of the original sequence. Depending on the value of the real number power, the resulting sequence may approach a finite limit, oscillate between two values, or continue to infinity. This information can be useful in various real-world applications, such as in modeling exponential growth or decay.

4. Can the limit of infinite sequence raised to a real number power be negative?

Yes, the limit of infinite sequence raised to a real number power can be negative. This can happen when the original sequence approaches a limit of zero or when the real number power is negative. In these cases, the resulting sequence will have alternating positive and negative values as it approaches the limit.

5. Are there any limitations to raising a limit of infinite sequence to a real number power?

Yes, there are some limitations to this process. The limit of the original sequence must exist for the limit of infinite sequence raised to a real number power to be meaningful. Additionally, the real number power must be defined for all values of the original sequence. For example, raising a sequence to a fractional or negative power may result in undefined values.

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