6–9. Determine whether the following sequences converge or div...

Question

6–9. Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.

{(−0.7)ⁿ}

Accepted Answer

Welcome back, everyone. Consider the sequence A subscript N equals -0.5 to the power of N. Number one determine whether the sequence a subscript and converges or diverges as an approaches infinity. So for part one, we can essentially notice that the sequence in the form of a subscript N equals -0.5. Ras the power of N is a geometric sequence because we have a constant race to the power of N, right? So, essentially what we have to do to answer this question is analyze the common ratio. Since a subscript N has a form of R to the power of N, where R is the common ratio, we want to analyze the magnitude of R. Our common ratio R is equal to 0.5. So we take the absolute value of -0.5, which is 0.5, and because it is less than 1, by definition, the geometric series converges, right? So the condition for the geometric series to converge is to have the absolute value of R less than 1, which is the case here. So we have our first answer for this problem. Number 2 says state whether the sequence is monotonic or oscillates. Let's recall that monotonic sequences. They either increase or decrease when an increases, right? They always keep increasing or always keep decreasing while sequences that oscillate generally oscillate between two values, right? So in this case, notice that a subscript N can be written as 0 0.5 to the power of N, which is the original form, and using the properties of exponents, we can also rewrite it as -1 to the power of N multiplied by 0.5 to the power of N. Notice that for even and values, -1 is going to be positive, and for odd and values, -1 to the power of N is going to be negative, right? So, essentially, our sequence oscillates between Positive and negative values, meaning for number 2 we can say that it oscillates. That's our answer for part 2, and for part 3, it says, if the sequence converges, find its limit. We're going to take a limit as N approaches infinity of a subscript N. Which is the limit as N approaches infinity of 0.5 to the power of N. And whenever we have a limit. In the form of limit. As n approaches infinity of art to the power of n. Where the absolute value of R is less than 1, such a limit is equal to 0, right, because we have a fractional term that is between -1 and 1 that is raised to an infinitely large exponent. So eventually. Our limit is going to be 0 or. This sequence because the absolute value of the common ratio is indeed less than 1, as we have shown in the first part of this problem. So the final answer for this problem is 0, that's the limit. Thank you for watching.

6–9. Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.

{(−0.7)ⁿ}

Key Concepts

Sequence Convergence and Divergence

Monotonicity of Sequences

Oscillating Sequences

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