Question
If the sequences {a } and (b } are divergent, then the sequence (a bn) Is divergent.
If the sequences {a } and (b } are divergent, then the sequence (a bn) Is divergent.
Answers
If $ \sum a_n $ and $ \sum b_n $ are both divergent, is $ \sum \left(a_n + b_n \right) $ necessarily divergent?
All that eventually. City, given that I am next divergent, some miss enough, and it's not equal to sleep off and consider being we energy sequence, which is equal to minus, and the submission of being will be equal to minus up some stuff in which is not equal to zero. So both the cities are divergent. Now when you take the submission of both the cities and that's being so that will be called to some is enough kn minus sale. So this is 4 to 0, and it is a convergence Siri's. So it did not necessarily true that there's a mission off. Two divergent series will also be divergent for the event statement for the answer to the Demonstrate Money's No.
Whose problem it helps to just write out the first few terms. So we know a one is one Hey, two is going to be four minus a one. So four minus one, which is three pay three is going to be four minus a too. So four minus three, which is going to be one. And from here, we know that the sequence should just continue in this way. One, three, one. But, you know, if you really want to convince yourself, you keep going. I thought I was going to be safe. Four minus a three is four minus one, which is three. A five is four minus a four four minus three, which is one. So this just switches between one and three, so that's definitely going to diverge. Okay. And once we got Tio Tio here, we knew that it was just going to switch between one and three, because each term is just defined only by the previous term. Okay, once you've gotten to hear, we've already been in this situation once before, right? So we know that if one is the previous term, that next term has to be three. So once we get to this one. We know that the next term has to be three. Okay, And then once we're at three, we know that the next term has to be one, because again, each term is defined only by the previous terms. So if we've been in the situation where we've gotten one before and the next term was three, we know that if one shows up again, the next time has to be three. And likewise, if three shows up in the next term is one, if that ever happens again and the three shows up, the next term is going to be one. But it's always good to, you know, you really convince yourself of it by, you know, right, Not a few more terms and seeing that indeed it is just going to alternate between one and three. So certainly divergence
And this problem let us first look at party party, we have Even equals to one. Mhm It too will be 4- Even which is 4 -3 for -1, which is three A three will be 4 -8, which is 4 -3, which is one. An evil Will be 4 -83 4 -1, which is three 85 will be four minus A. For which is four minus three. Which is what? So what we observe is that the sequence Mhm Oscillates between one and 3, which means that limit and tends to infinity A. N does not exist, which means that these sequences diversion. Let us look at part B. We have even equals to two. A two equals to 4 -7, is just 4 -2, which is two, It is four minutes later which is again 4 -2 which is two. And similarly, if results go to if I will also so we see itself constant sequence, which means that will limit and tends to infinity and exists and it is equal to two which means this is convergent, that's all