Use the Alternating Series Test, if applicable, to determine...

Use the Alternating Series Test, if applicable, to determine the convergence or diver $$sum_{n = 3}^{infty} frac{(-1)^{n}n}{n^{2} - 8}$$ Identify a_n. Evaluate the following limit. lim_{n o infty} a_n Since lim_{n o infty} a_n ? 0 and a_{n + 1} ? a_n for all n, ---Select---.

Transcript

00:01 So in the given question we need to use the alternating series test if applicable in order to determine the convergence or divergence of the given series which is minus of 1 power of n in multiplication with n divided by n square minus of 8.

00:20 Okay, so as per the alternating series test firstly we need to know what are the conditions which we need to fulfill in order to show that the given series is convergent or divergent.

00:31 So we'll write the alternating series test.

00:37 Okay, so for this we'll write sigma n varies from 1 to infinity this is minus of 1 power of n minus 1 this is b n which is equal to b 1 minus of b 2 plus of b 3 and it will just go on like this where b is greater than 0.

01:00 So this will be satisfying two conditions.

01:03 So you'll write satisfied firstly b n plus 1 is less than equals to b n for all n and secondly we have limit of n tending to infinity and this is b n which is equals to 0.

01:25 So in that case we say that the series is convergent.

01:28 Right, then the series is convergent.

01:35 Okay, fine.

01:37 Okay, so you can take either b n and instead of b n you can also write it as a n it doesn't matter don't worry about that.

01:46 So here the value of b n is n divided by n square minus of 8 and this value will be greater than 0 for n greater than equals to 3.

01:58 Yes, if you will put 2 you will get 2 square 4 minus 8 which will be negative.

02:03 That is i have written greater than equals to 3.

02:05 So if i'm starting with 3 this means that this is my initial value which is b 0 b 1 b 2 like this.

02:12 So if i start with n is equals to 3 and i write n is equal to 4 then i write n is equal to 5 like this.

02:21 So i'll be getting this value as 3 by 1 comma 4 by 8 comma 5 by 17 and this is comma 6 by 28.

02:33 Okay, now you can understand this point that as soon as you're increasing the n's the denominator is becoming larger.

02:41 So when the denominator is larger the fraction is small.

02:45 So this clearly means that 6 by 28 will be less than 5 by 17.

02:51 Now what does this indicate? this indicate that b n plus 1 is less than b n.

02:57 Alright, i hope you got the answer for this...

Question

Use the Alternating Series Test, if applicable, to determine the convergence or diver $$sum_{n = 3}^{infty} frac{(-1)^{n}n}{n^{2} - 8}$$ Identify a_n. Evaluate the following limit. lim_{n o infty} a_n Since lim_{n o infty} a_n ? 0 and a_{n + 1} ? a_n for all n, ---Select---.

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