Does the series
∑ (from n=1 to ∞) (1/n − 1/n²)
con...

Question

Does the series

∑ (from n=1 to ∞) (1/n − 1/n²)

converge or diverge? Justify your answer.

Accepted Answer

Hello there, today we are going 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. Determine whether the series, the sum evaluated from n equals 1 to infinity of 1 divided by n minus 1 divided by n to the power of 3 divided by 2 converges or diverges. OK, so it appears for this particular prom we're asked to take the series that is provided to us, and we're asked to determine whether or not it will A converge or B diverges. So now that we know that we're ultimately trying to determine whether or not this series converges or diverges. Our first step that we need to take is we need to let ann be equal to 1 divided by n minus 1 divided by n to the power of 3 divided by 2. Fantastic. So with that in mind, we also need to note that we will have. A N is greater than 0, and because A N is greater than 0, we need to note that we can recall and use comparison tests for positive term series. So with that in mind, we now need to recall and use the limit comparison test. And based on our prior knowledge, we can recall that the limit comparison test states that if we let a n be greater than 0 and bn be greater than 0 for all lowercase n is greater than or equal to N, where N is an integer. And we assume the limit L is equal to the limit as N approaches infinity of A and divided by BN exists in square bracket 0 infinity parenthesis. And if 0 is less than l and l is less than infinity, then the sum of a n and the sum of bn either both converge or both diverge. And if l is equal to zero and the sum of bn converges, then the sum of a n converges. And if L is equal to infinity, and the sum of BN diverges, then the sum of A N diverges. Otherwise, the test will be inconclusive. So with that in mind, we now need to compare a n with the harmonic series which states that the sum evaluated from n equals 1 to infinity of 1 divided by n, and we're comparingnn to this harmonic series and we need to note that the sum evaluated from n equals 1 to infinity of 1 divided by n. Is a divergent P series, so that means that this is the case since P is equal to 1 and 1 is less than or equal to 1. Fantastic. So with that in mind, we need to note that we can, so based with this, so since we're armed with this information and based on this information, we can now compute the limit. So we can go ahead and write the limit as n approaches infinity of 1 divided by n minus 1 divided by n to the power of 3 divided by 2, all divided by 1 divided by n. Will be equal to the limit as n approaches infinity of lowercase n multiplied by parentheses of 1 divided by n minus 1 divided by n to the power of 3 divided by 2. Which will be equal to the limit as n approaches infinity of 1 minus 1 divided by n to the power of 1/2. Which will be equal to square brackets of 1 minus 1 divided by infinity to the power of 1/2, which will be equal to 1. And since 0 is less than L and L is equal to 1 is less than infinity. And that the sum evaluated from n equals 1 to infinity of 1 divided by n diverges, so I put a d in quotes, we can safely say that the sum evaluated from n equals 1 to infinity of parentheses of 1 divided by n minus 1 divided by n to the power of 3 divided by 2 will also. Diverge. So it also diverges, and that's it. We've solved for our final answer. Hooray, we did it. So going back to the top to look at our multiple choice answers, we could safely say that our final answer without a doubt will be B diverges, and that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Does the series∑ (from n=1 to ∞) (1/n − 1/n²)converge or diverge? Justify your answer.

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Does the series
∑ (from n=1 to ∞) (1/n − 1/n²)
converge or diverge? Justify your answer.