9–36. Comparison tests Use the Comparison Test or the Limit Co...

Question

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.

∑ (k = 1 to ∞) (2 + (−1)ᵏ) / k²

Accepted Answer

Hello there. Today we want to 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 1 to positive infinity of 1 parentheses -1 to the power of M divided by the power of 2 converge or diverge. Use the direct comparison test or limit comparison test. Awesome. So it appears for this particular problem we're asked to take the series that is provided to us by the problem itself, and we're asked to determine whether or not this particular series converges or diverges using the direct comparison test or the limit comparison test. So that will mean our final answer will be either a converges or B diverges. So with that in mind, our first step that we need to take is we need to simplify and observe the terms that are provided to us. So when we simplify and observe the terms that are given to us, we will need to note that parentheses -1 to the power of M will be equal to curly bracket of -1. And, and it'll be -1 if M is an odd value and 1 if M is an even number or even value. So that will mean that the numerator 1 + parentheses -1 to the power of M will equal 0 when M is odd. And it will be 2 when M is even. Fantastic. Therefore, only even values of M contribute non-zero terms, so that will mean as a result, the series is equivalent to summing the even index terms, meaning. The sum evaluated at M equals 1 infinity of 1 + -1 to the power of M divided by m2 will be equal to the sum evaluated at M equals 2 to positive infinity, where we need to note that M equals 2, where M is even. Of 2 divided by M to the power of 2, we need to write that M is equal to 2 multiplied for so we need to note that M is equal to 2 multiplied by K for even indices, meaning, let's make a note of this meaning. Put a star k is equal to 123, etc. Then that will mean that 2 divided by m2 will be equal to 2 divided by parentheses 2 multiplied by k 2, which is equal to 2 divided by 4 multiplied by k to the power of 2, which will be equal to 1 divided by 2 multiplied by K to the power of 2 when we simplify. So, moving right along here. So that will mean our series will become. The sum evaluated from K equals 1 to positive infinity of 1 divided by 2 multiplied by k 2, which will be equal to 1/2 multiplied by the sum evaluated from K equals 1 to positive infinity of 1 divided by K to the power of 2. We need to be able to recognize at this point that the series will diverge according to the P test as P is equal to 2. So now our second step is we need to recall and use the convergence test, and by using the direct comparison test for every m value, that will mean that 0 is less than or equal to 1 + -1 to the power of divided by m2, which is less than or equal to 2 divided by m2. And we need to note that the sum of 2 divided by N to the power 2 converges. So we'll make a note so see in quotation marks for converges, and because it converges, that will mean that the original series will converge as well. So that means that our final answer without a doubt is converges. Hooray, we did it. So looking at our multiple choice answers, the correct answer has to be, without a doubt, multiple choice answer letter A, which once again is converges. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.

∑ (k = 1 to ∞) (2 + (−1)ᵏ) / k²

Key Concepts

Comparison Test

Limit Comparison Test

Behavior of Alternating Terms in Series

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