Goodreads helps you follow your favorite authors. Be the first to learn about new releases!
Start by following James Stewart.
Showing 1-7 of 7
“You know, I just love Grace Kelly. Not because she was a princess, not because she was an actress, not because she was my friend, but because she was just about the nicest lady I ever met. Grace brought into my life as she brought into yours, a soft, warm light every time I saw her, and every time I saw her was a holiday of its own. No question, I’ll miss her, we’ll all miss her, God bless you, Princess Grace.”
―
―
“Behind Calvary's cross is the throne of heaven.”
―
―
“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”
― Calculus: Early Transcendentals
, 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”
― Calculus: Early Transcendentals
“I hate that I have to spend half my life doing a job I hate for a boss I can't stand.'
'Haha, yeah, we live in a society.'
'The point isn't that we live in it, but that we built it. And we could just have easily built it differently.”
― Dinosaur Therapy
'Haha, yeah, we live in a society.'
'The point isn't that we live in it, but that we built it. And we could just have easily built it differently.”
― Dinosaur Therapy
“I'm sad.'
'I'm sorry and I'm here for you.'
'Aren't you going to tell me to cheer up? People always tell me to cheer up.'
'No, I still like you when you're sad.”
― Dinosaur Therapy
'I'm sorry and I'm here for you.'
'Aren't you going to tell me to cheer up? People always tell me to cheer up.'
'No, I still like you when you're sad.”
― Dinosaur Therapy
“He glanced at Artison. “They couldn’t even kill you if they tried.” “What are you talking about? Why not?” “Because,” Larria whispered, “You’re already dead.”
― The Dreamers' Relics: Dream World Trilogy
― The Dreamers' Relics: Dream World Trilogy
“I'm useless. I quit everything I try.'
'That doesn't make you useless. It means you haven't quit on finding the thing that's right for you.”
― Dinosaur Therapy
'That doesn't make you useless. It means you haven't quit on finding the thing that's right for you.”
― Dinosaur Therapy
