I'm on Question 89

nasi112

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n=0(m=0xm)n\sum_{n = 0}^{\infty} \left(\sum_{m=0}^{\infty} x^{m}\right)^n
This question is connected to question 88 from that video. blackpenredpen explained all 88 questions well, but this time I'm completely confused.

Question 88 n=0(11x)n\sum_{n = 0}^{\infty} \left(\frac{1}{1 - x}\right)^n
Are questions 88 and 89 essentially the same? If they are, why do they converge on different domains for x?
 
They are not the same. The first one is defined everywhere, the second one has a singularity at x=1. x=1.

The first one requires x<1 |x|<1 to have even a chance to converge. But there could be additional requirements; I haven't computed the limit for those values.

Every infinite series k=0an \displaystyle{\sum_{k=0}^\infty }a_n can only converge, if limnan=0. \displaystyle{\lim_{n \to \infty}a_n=0}.
This is a necessary condition, however, not a sufficient one as an=1n a_n=\dfrac{1}{n} shows. Anyway, we need
limn(11x)n=0, \displaystyle{\lim_{n \to \infty}\left(\dfrac{1}{1-x}\right)^n}=0, which is not the case for 0x<1. 0\leq x<1.
 
n=0(m=0xm)n\sum_{n = 0}^{\infty} \left(\sum_{m=0}^{\infty} x^{m}\right)^n
This question is connected to question 88 from that video. blackpenredpen explained all 88 questions well, but this time I'm completely confused.

Question 88 n=0(11x)n\sum_{n = 0}^{\infty} \left(\frac{1}{1 - x}\right)^n
Are questions 88 and 89 essentially the same? If they are, why do they converge on different domains for x?
Thanks for showing more; the link is

1750352796858.png
1750352857579.png

These are the same apart from domain issues. I think he explains it quite well. Please tell us more of what makes you unsure.

He first transforms 89 to the form 88, which can be done as long as x is in -1 < x < 1. This is necessary so that the terms of the outer summation in 89 are even defined.

In turn, 89 converges as long as x is in the domain of 88, namely x < 0 or x > 2.

The domain of 89 is therefore the intersection of the two, -1 < x < 0.

Frankly, I have taken too long trying to make sure I say this correctly. I hope I have it right. I really don't like explaining videos, especially when they are hard to step back and forth through, like this very long one!
 
Reply to the fresh_42

When I solved the outer sum of question 89, I didn’t use the limit limn(11x)n\lim_{n\to \infty}\left(\frac{1}{1 - x}\right)^n because I recognized right away that the sum was a geometric series. However, after seeing your approach, I think it would be worthwhile to try starting with the limit next time. If I started with that limit first, I probably wouldn’t have thought to consider the absolute value. I would approach it like this: limn(11x)n=(11x)\lim_{n\to \infty}\left(\frac{1}{1 - x}\right)^n = \left(\frac{1}{1 - x}\right)^{\infty}, and then I’d conclude that this limit doesn’t go to infinity when (11x)1\left(\frac{1}{1 - x}\right) \leq 1. Since x = 1 is not in the domain, I’d exclude it, leading me to x<0x < 0 but I will miss x>2\displaystyle x > 2 because I didn't consider the absolute value. So yes, I agree that starting with the limit method could partially provide a strong foundation for solving the question.


Reply to the Doctor

After thinking carefully about what you said, I believe I now understand most of the idea. There are two domains to consider: I know the inner domain is (1,1)(-1 , 1), and the outer domain is (,0)(2,)(-\infty, 0) \cup (2, \infty) . If I understand correctly, I need to find the values that belong to both domains, in other words, values that make both the inner and outer series converge. The interval (2,)(2, \infty) must be excluded because it causes the inner series to diverge. Similarly, (0,1)(0, 1) must be excluded because it makes the outer series diverge. That leaves (1,0)(-1,0) where both series converge. So essentially, what I did was take the intersection of the two domains, even though I didn’t fully realize it at first, it only became clear to me after working through the process.


Reply to the blamocur

If my explanation to the Doctor was correct, the values of x would be in this domain (1,0)(-1,0)
 
If my explanation to the Doctor was correct, the values of x would be in this domain (−1,0)(-1,0)(−1,0)
Your answer to @Dr.Peterson looks good to me, but I believe the answer to my question, (i.e., for which values of xx questions 88 and 89 are equivalent), is what you call "inner domain", i.e, (1,1)(-1,1).
 
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