Sum 2 Prove Quotes

Incomplete,” he says. “If I’m whole, why do I feel like I’m not?” And as usual, Roberta has a calming platitude intended to ease his mind, but as time goes on her rote wisdom leaves him flat and disappointed. “Wholeness comes from creating experiences that are solely yours, Cam,” she tells him. “Live your life and soon you’ll find the lives of those who came before won’t matter. Those who gave rise to you mean nothing compared to what you are.” But how can he live his life when he’s not convinced he has one? The attacks in the press conference still plague him. If a human being has a soul, then where is his? And if the human soul is indivisible, then how can his be the sum of the parts of all the kids who gave rise to him? He’s not one of them, he’s not all of them, so who is he? His questions make Roberta impatient. “I’m sorry,” she tells him, “but I don’t deal in the unanswerable.” “So you don’t believe in souls?” Cam asks her. “I didn’t say that, but I don’t try to answer things that don’t have tangible data. If people have souls, then you must have one, proved by the mere fact that you’re alive.” “But what if there is no ‘I’ inside me? What if I’m just flesh going through the motions, with nothing inside?” Roberta considers this, or at least pretends to. “Well, if that were the case, I doubt you’d be asking these questions.” She thinks for a moment. “If you must have a construct, then think of it this way: Whether consciousness is implanted in us by something divine, or whether it is created by the efforts of our brains, the end result is the same. We are.” “Until we are not,” Cam adds. Roberta nods. “Yes, until we are not.” And she leaves him with none of his questions answered.

Notice that if , then and , whereas if , then and . (a) If is absolutely convergent, show that both of the series and are convergent. (b) If is conditionally convergent, show that both of the series and are divergent. 44. Prove that if is a conditionally convergent series and is any real number, then there is a rearrangement of whose sum is . [Hints: Use the notation of Exercise 43. an an 0 a n an 0 an an 0 an 0 an an an a n an an a n an r an r Take just enough positive terms so that their sum is greater than . Then add just enough negative terms so that the cumulative sum is less than . Continue in this manner and use Theorem 11.2.6.] 45. Suppose the series is conditionally convergent. (a) Prove that the series is divergent. (b) Conditional convergence of is not enough to determine whether is convergent. Show this by giving an example of a conditionally convergent series such that converges and an example where diverges. r an r an n 2 an an nan nan nan an We now have several ways of testing a series for convergence

Oresme's genius was to make a new series which was definitely smaller than the harmonic series. He took the list of all unit fractions, and for any of them which did not have a power of two as a denominator, he replaced it with a smaller fraction which did. As all these new fractions were either the same or smaller, the total of this new series had therefore to be smaller than the sum of the harmonic series. But when Oresme grouped these fractions into runs, each of which added up to 1/2, he was left with a sum of an infinite sequence of 1/2s, which definitely diverges. This meant in turn that the greater harmonic series must also diverge. Oresme had proved that a sequence of ever-decreasing numbers could still be divergent. (His proof was lost for a while, and the same result was independently rediscovered in the 1600s.)

If you recall, in Ramanujan's letters to English mathematicians, he claimed that 1 + 2 + 3 +...= -1/12. He was so surprised when Hardy took him seriously that he replied on 27 February 1913 in the following words: 'I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study Infinite Series and not fall into the pitfalls of divergent series. If I had given you my methods of proof I am sure you will follow the London Professor. I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 +...= -1/12 under mu theory. If I tell you this, you will at once point out to me the lunatic asylum as my goal.' It turns out that not only had Ramanujan independently rediscovered the Bernoulli numbers, but he may have found more than one way to prove that 1 +2 +3 + 4 ...= -1/12. This is now called Ramanujan summation and gives us an insight into the ways in which the sum of a sequence can be divergent. Of course, the sum of all the positive whole numbers is infinite, but if you can somehow peel that infinity back out of the way and look at what else is going on, there's a -1/12 in there.