Sia

“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”
― James Stewart, Calculus: Early Transcendentals

“all started at the Temple of Apollo In Delphi. One of his friends approached the oracle with the question: “Is anyone wiser than Socrates?” the answer was “No.” Socrates was profoundly puzzled by this episode. He claimed to know”
― Plato, The Socratic Dialogues

“Hot, isn’t it?” Peter asked on the way into town, and the driver nodded. He could hear from the accent in his French that he was American, but he spoke it adequately, and the driver answered him in French, speaking slowly, so Peter could understand him. “It’s been nice for a week. Did you come from America?” the driver asked with interest. People responded to Peter that way, they were drawn to him, even if they normally wouldn’t have been. But the fact that he spoke French to him impressed the driver.”
― Danielle Steel, Five Days in Paris

“Jobs had a tougher time navigating the controversies over Apple’s desire to keep tight control over which apps could be downloaded onto the iPhone and iPad. Guarding against apps that contained viruses or violated the user’s privacy made sense; preventing apps that took users to other websites to buy subscriptions, rather than doing it through the iTunes Store, at least had a business rationale. But Jobs and his team went further: They decided to ban any app that defamed people, might be politically explosive, or was deemed by Apple’s censors to be pornographic.”
― Walter Isaacson, Steve Jobs