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Calculus and Beyond Homework Help
Converging Sequence: Basic Steps and Practice Problems
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[QUOTE="pasmith, post: 4519788, member: 415692"] There are various techniques for limit problems, but since this problem asks you to start with an arbitrary convergent sequence [itex](a_k)[/itex] the only one which will work is to use what you know about [itex](a_k)[/itex]: for all [itex]\epsilon > 0[/itex] there exists [itex]K \in \mathbb{N}[/itex] such that if [itex]k \geq K[/itex] then [itex]a - \epsilon < a_k < a + \epsilon[/itex]. That suggests taking an arbitrary [itex]\epsilon > 0[/itex] and its corresponding [itex]K[/itex] and splitting the sum as follows: [tex] \frac1n \sum_{k=1}^n a_k = \frac1n \sum_{k=1}^{K-1} a_k + \frac1n \sum_{k=K}^n a_k [/tex] (You are interested in the limit [itex]n \to \infty[/itex], so at some stage you will have [itex]n > K[/itex] and you may as well assume that to start with.) Your plan is to show that [tex] a - \epsilon \leq \lim_{n \to \infty} \frac1n \sum_{k=1}^n a_k \leq a + \epsilon [/tex] and since [itex]\epsilon > 0[/itex] was arbitrary it must follow that [tex] \lim_{n \to \infty} \frac1n \sum_{k=1}^n a_k = a [/tex] as required. [/QUOTE]
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Converging Sequence: Basic Steps and Practice Problems
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