Consider the sequence (1)/(3), (1)/(9), (1)/(27), (1)/(81), …, (1)/(3^1), …(a) What happens to t

step 1 First problem we want to consider the sequence 1 3rd, 1 9th, 127th, 181st, and continuing down,

Consider the sequence (1)/(3), (1)/(9), (1)/(27), (1)/(81),...

Consider the sequence $\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \ldots, \frac{1}{3^{1}}, \ldots$ (a) What happens to the terms of this sequence as $k$ gets larger and larger? Express your answer in limit notation. (b) Find a sufficiently large value of $k$ so that every term past the $k$ th term of this sequence will be less than $0.0001 .$

Consider the sequence $\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \ldots, \frac{1}{3^{1}}, \ldots$
(a) What happens to the terms of this sequence as $k$ gets larger and larger? Express your answer in limit notation.
(b) Find a sufficiently large value of $k$ so that every term past the $k$ th term of this sequence will be less than $0.0001 .$

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Key Concepts

Limits of Sequences

This concept is concerned with the behavior of a sequence as its index approaches infinity. It involves determining whether the terms approach a finite value (convergence), diverge to infinity, or oscillate without settling. Expressing this behavior in limit notation is a fundamental part of understanding sequences in calculus.

Geometric Sequences

A geometric sequence is one in which each term is derived by multiplying the previous term by a constant called the common ratio. This structure allows for predictable behavior, especially when the common ratio is less than 1 in absolute value, which leads to convergence towards zero as the sequence progresses.

Exponential Decay

Exponential decay describes the process where quantities decrease at a rate proportional to their current value. In sequences with a common ratio less than 1, the terms decrease rapidly, and this concept is used to analyze how quickly the terms approach zero, making it easier to determine when they become smaller than a given threshold.

Inequality Solving with Exponentials

This idea involves finding the point at which the terms of an exponentially decaying sequence fall below a certain value. Techniques often include taking logarithms to solve for the index at which the inequality holds, which is crucial for establishing bounds in sequence analysis.

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