11–86. Applying convergence tests Determine whether the follow...

Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞) 2ᵏ / (3ᵏ − 2ᵏ)

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Does the series, the sum, Evaluated from M equals 12 positive infinity of 3 to the power of divided by 6 to the power of m minus 3 of M converge or diverge. Awesome. So it appears for this particular prom we're asked to take the series that is provided to us by the prom itself, and we're asked to determine whether it's a, yes, the series converges, or B, no, the series diverges. So now that we know what we're ultimately trying to solve for, our first step that we need to take is we need to simplify the general term. Meaning, we need to factor out 3 power of M out of the denominator, so we need to take 3 power of M divided by 6 M minus 3 power of M. Which is going to be equal to 3 of M divided by 3 of M multiplied by parentheses 2 M minus 1, which when we simplify, will be equal to 1 divided by 2 power of M minus 1. So the series will become the sum, so once again we'll have our series become the following. The sum evaluated from M equals 1 to positive infinity of 1 divided by 2 to the power of M minus 1. Fantastic. Our second step now is we need to compare our series to a geometric series, and we need to recall a note that for every integer M is greater than or equal to 1. We will have that 2 to the power of m minus 1 will be greater than or equal to 2 to the power of m minus 1. Because 2 of M minus 1 minus 2 M minus 1 is equal to 2 of M minus 1, -1 is greater than or equal to 0. Which this is indeed true for the condition where 2 to the power of m minus 1 is greater than or equal to 1 for M is greater than or equal to 1. So therefore we could go ahead and write that 1 divided by 2 to the power of m minus 1 is less than or equal to 1 divided by 2 power of M minus 1. Which will be equal to 2 multiplied by 1 divided by 2 to the power of M. Fantastic. So that will mean that our 3rd step, so step number 3, we need to recall and use the comparison test. So we need to recall and note that the geometric series, which is the sum evaluated from M equals 1 to positive infinity of 1 divided by 2 to the power of M. Converges, which I'll simply call C, since its sum is 1. And since for every m we can state that 0 is less than or equal to 1 divided by 2 power of M minus 1 is less than or equal to 2 multiplied by 1 divided by 2 to the power of M. So based on this information, by the comparison test, the series, the sum, once again, our series, the sum of Drum roll, so we have our series. The sum evaluated from M equals 1 to positive infinity of 1 divided by. 2 multiplied, so it's gonna be 2 to the power of M minus 1 also converges. So that means that our final answer, without a doubt, will have to be multiple choice answer letter A. Which is yes, because the comparison test also confirms that this series is indeed converging. So our final answer once again is A, yes, the series converges. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) 2ᵏ / (3ᵏ − 2ᵏ)

Key Concepts

Infinite Series and Convergence

Comparison Test

Ratio Test

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