Question

I've been working on a problem that is asking for intervals of convergence. It's actually, 4-in-1, and if I don't get them all right, I don't get any right. I was pretty sure I had the first three, but the last interval (that I keep getting) isn't even available as a choice. Screenshot attached. Am I missing something obvious? I'll probably check in the morning. Thanks!

Answer 1

What's the original problem given?

Answer 2

They're looking for the intervals of convergence for each of the four sums I have highlighted in blue.

Answer 3

Ok, I'll work on those power series right now

Answer 4

Thanks! Here's a screenshot from the actual prob.

Answer 5

Embarrasingly, I've made 14 attempts now. :/ I think I'm sticking with it more out of spite.

Answer 6

that helps

Answer 7

Ok solved the first one... The first one is a geometric series of the form

\[\sum_{n=0}^{\infty} a.r ^{n}\]

\[r=\frac{ x-6 }{ 6 }\]

For a geometric series the interval is -1<r<1, so

\[-1< \frac{ x-6 }{ 6 } < 1\]
\[-6< x-6 < 6\]
\[0< x < 12\]

I= 0 - 12 (not using the brackets here cause I haven't checked whether 0 and 12 are included or not.

So answer C is the correct answer for question 1.

Answer 8

It seems to me you calculated that correctly, but still wrote the wrong letter in front of question 1.

Answer 9

Second exercise... Here you have to use the ratio test in order to get rid of the factorial. If you see a factorial and have to do a limit, use the ratio test.

Ratio test:
\[\lim_{n \rightarrow \infty} \left| \frac{ c_{n+1} }{ c _{n} }\right|\]

So you add +1 with every n of the series and multiply with the inverse of the original

\[\lim_{n \rightarrow \infty}\left| \frac{ (x-6)^{n+1} }{ (n+1)!6^{n+1} } . \frac{ n! 6^{n} }{(x-6)^{n} ? }\right|\]

reorder it for clarity

\[\lim_{n \rightarrow \infty}\left| \frac{ (x-6)^{n+1} }{(x-6)^{n} } . \frac{ n! } {(n+1)! } \frac{ 6^{n} }{ 6^{n+1}}\right|\]

You already know what to do with the exponents.

\[\lim_{n \rightarrow \infty}\left| \frac{ (x-6)^{1} }{1 } . \frac{ n! } {(n+1)! } \frac{ 1 }{ 6^{1}}\right|\]

That is one of the reasons you use the ratio test because the limit of number ^(infinity) is undefined, so you want to get rid of the n-exponents.

If you divide a factorial by a factorial that's one higher only the last number remains: example

\[\frac{ 4! }{ 5! }=\frac{ 4.3.2.1 }{ 5.4.3.2.1 } = \frac{ 1 }{ 5 }\]

So you can also get rid of the factorial


\[\lim_{n \rightarrow \infty}\left| \frac{ (x-6)} {(n+1) 6}\right|\]

get the constants out

\[\left| \frac{ x-6 }{ 6 }\right| \lim_{n \rightarrow \infty}\left| \frac{ 1 }{ n+1 }\right|\]

the limit of 1/infinity = 0 so

\[\left| \frac{ x-6 }{ 6 }\right|.0 < 1\]

in order to converge. Since you multiply by 0 it doesn't matter what x has for a value. x can be any number you want.

\[x \in R \]

so the interval is (-infnty, + infnty). Question 2 has A for an answer.

Answer 10

For problem 3, again use the ratio test. You'll get

\[\left| 7x \right| \lim_{n \rightarrow \infty} \left| \frac{ n ^{6} }{ (n+1)^{6} } \right|\]

I'm not gonna bother with calculating (n+1)^6, or applying L'Hospital's rule. I can see that the highest power for n is 6 in the denominator and the numenator. So I know that the limit will be the divide of the coefficients of those n with the highest degree, which is 1/1 = 1. So

\[\left| 7x \right|.1 <1\]
or
\[-1<7x<1\]
\[-\frac{ 1 }{ 7 }<x<\frac{ 1 }{ 7 }\]

So the I = from -1/7 to 1/7 and you answered that correctly

Answer 11

The last one... using the ratio testyou'll discover after canceling out the factorials that you're still left with a lim (n+1), which is infinity and always > 1. So the series actually diverges. The whole series can only converge if it's 0. That can only happen if (7x-6)/6 = 0. The singular point then for x = 6/7.

Answer 12

If you want to find out more tests to determine intervals, radius and convergence/divergence I advize you this youtube channel: https://www.youtube.com/user/patrickJMT/videos?shelf_id=4&view=0&sort=dd

He has many examples and sets up a strategy plan to attack the different types of series, aside from the ratio and root test.

Answer 13

BTW that same youtube channel helped me immensely in getting to set up taylor series. Some are really challenging, like the one I posted about yesterday. I didn't even know about double factorials until yesterday. Sometimes seeing the pattern requires time and playing around with numbers. But I do find it very interesting to figure out what the heck is going on if you take the primes, and to learn to set up general functions. It's a necessity for physics. Right now I'm doing exercises on the Frobenius method (for DE with singularities), tomorrow I'll start on systems of DEs.

Answer 14

Seriously, thank you *so* much! I originally had the first one right, but had gotten so flustered by Webwork not telling me exactly *which* piece that I was missing, that I started flailing around when I couldn't get all four right. I definitely never would have gotten the last one without your help, though.

Also, it's good to hear from someone else studying physics who knows what this is all for. Sometimes the math department gets so far from the reasons for the math, I feel like I'm just doing Sudoku puzzles for the severely masochistic. (And I *love* math).

I'm a big fan of PatrickJMT, I'll have to check the link. Again, thanks. I may survive through my honors physics class next semester after all! :)

Answer 15

Hmm, even the physic subjects are fairly abstract and theoretical math. The actual math subjects leave out context, But the physics subject parallel or in series with it implements and refers to the math constantly. Actually, to me the physics subjects are math subjects too and remain mathematically theoretical. They just put it in a physical analytical context. I can't really say they "apply" the math, because so far I've only had very few subject where I had to use a calculator - either general subjects or statistics or lab. For almost 3/4 we never have to use a calculator (and the professors say it with pride- calculating stuff? That's for the engineers). It's all analytical proof and setting up theoretical expressions. I don't mind that though. I already know the engineering part of physics with my previous studies of industrial designer. Both for career reasons and motivation I felt I was ready to for the purely theoretical part of it, and I greatly enjoy figuring it out. Sometimes it feels equal to the brain-buzz you can get during meditation. But it needs time. I have 5 exams by the end of August. This math 3 course, statistics, analytical mechanics, biophysics and classic field theory. I've noticed that I make more headway when I'm stumped over a problem to set it aside for a while and pick another subject to review, and then when I come back to it I might have an inkling of an idea in which direction to search further.

BTW problem 4: I was stumped at first as well, because I thought... how does that even have a convergence solution when it's a diverging series. Then I recognized the solution given was a singular point, and with the singular point at least the series converges to zero. So the convergence interval can only be that singular point. I hadn't come across such a problem yet, so I learned and figured it out as I tried to help you. So, I learned too and thank you for posting the problem!

Answer 16

Ah, that makes sense on number 4. I'm glad the exercise was useful for you as well.

We don't get to use calculators for this course. It's funny, because I still have the HP 48-g calculator that the same university *made* me buy for Calc I over 20 years ago. (I passed the course just fine, but they made me retake it because of how long ago it was).

I'm noticing that it takes more discipline for me to put down a tough problem or ask for help than just about anything else. I will say, though, that I've been extremely disappointed with our book, online homework, and the recorded lectures, especially for this section. I don't think it would be anywhere near so difficult for students if they were to spend a little more time on the individual pieces in order to develop an intuition for the tougher problems progressively. It's as though I've been asked to eat an entire pie in one sitting!