Question 1 Determine whether each of the following series...

Question 1 Determine whether each of the following series converges absolutely, converges conditionally, or diverges. (a) \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{3n + 1} (b) \sum_{n=1}^{\infty} \frac{\cos n}{n^2 + 9}

Transcript

00:01 In the part a of the problem, the given series is summation of n equals to 1 up to infinity minus 1 raised to the power of n which is divided by 5 multiplied with n.

00:17 So we can write this series as what? as 1 divided by 5 multiplied with n equals to 1 up to infinity minus 1 raised to the index of n divided by n.

00:32 So what we will do, we will apply the alternating series test.

00:42 So according to the alternating series test, we write that what an should be what? be positive.

00:55 And what and the limit of n tends to infinity over a n should be what? should be equals to 0.

01:08 So now therefore in this particular series, we see that what that limit n tends to infinity over 1 divided by n equals to what? equals to 0.

01:23 That is what? that is it converges.

01:27 Alright, so we see that there can be two conditions over here.

01:33 That is, it can be either absolutely convergent or it can be what? it can be conditional convergence.

01:41 Alright, it can be conditional convergence.

01:44 Alright, so now we see that summation of a n is what is absolutely convergent.

01:51 It is absolutely convergent when summation of modulus of a n what? is convergent.

02:02 Alright, so therefore we write that a n is conditionally.

02:08 We write that summation of a n is conditionally convergent or we just simply write conditional when what? when summation of what? of modulus of a n is what? is divergent.

02:26 Is divergent and the other condition is that and the summation of simple a n is what? it is convergent...

Question

Question 1 Determine whether each of the following series converges absolutely, converges conditionally, or diverges. (a) \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{3n + 1} (b) \sum_{n=1}^{\infty} \frac{\cos n}{n^2 + 9}

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