Determining Convergence of Sequences
Which of the sequen...

Question

Determining Convergence of Sequences

Which of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = 1 + (0.9)ⁿ

Accepted Answer

Welcome back everyone. Consider the sequence ann defined by the general term ann equals 2 minus 4/5 to the power of n. Determine whether the sequence converges or diverges. If it converges, identify its limit l. So for this problem we're going to apply the nth term test. Remember that we want to evaluate the limits and approaches infinity of a n, and in this case that would be limits and approaches infinity of 2 minus 4/5 raised to the power of n. We can split our limit into two parts, that be limitsn approaches infinity of 2 minus limitsn approaches infinity of 4/5 to the power of n. So for the first limit we have the limit of a constant, which is equal to that constant itself and that's 2. And for the second limit, let's notice that we have our base 4/5. The absolute value of 4/5 is less than 1. And whenever this is the case and we are raising it to an infinitely large exponent, the limit is basically 0, so we can cross out the second limit and show that our limit is 2 minus 0, which is equal to 2. Notice that this limit is finite, and whenever this is the case, our sequence converges. We're going to say that the sequence converges, and it converges to the value of the limit. In particular, our limit l is equal to 2. Let's label our final answer and thank you for watching.

Determining Convergence of SequencesWhich of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.aₙ = 1 + (0.9)ⁿ

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Determining Convergence of Sequences
Which of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = 1 + (0.9)ⁿ