Y!A: What is the Taylor series of ln((x^2)+1) centered at c=0?

this is a question i answered on Y!Answers and can be found here:

http://answers.yahoo.com/question/index;_ylt=AmsLM3qKuCKKGf8_wa161Zfty6IX;_ylv=3?qid=20090426105041AA1f5lh&show=7#profile-info-2AVjODBNaa

there are two ways you can do this:

the first is by deriving the taylor series directly. however that involves a lot of messy product rules for derivatives in this case (but i will start it for you anyway).

the second is to use substitution and integration.

THE FIRST METHOD:
you would write normally, as if you were deriving a taylor series:
f(x) = a ₀ + a₁x + a₂x² + ... = ln(x² + 1)

but you can see that when we differentiate both sides, ln(x² +1) yields confusing derivatives. so, i will use a different, smarter method.

THE SECOND METHOD:
the first thing i will do is take a known taylor series, and later manipulate it:
the taylor series for f(x) = 1/(1-x) is known to be 1 + x + x² + x³ + x⁴ + ...
so i will take f(-x) = 1 / (1 - (-x) ) = 1 / (1+x)
for which the taylor series becomes: 1 - x + x² - x³ + x⁴ - ... (plugging in -x everywhere for x)

next, i integrate 1 / (1+x) and its taylor series:
ln(1+x) = C + x - (x²)/2 + (x³)/3 - (x⁴)/4 + (x⁵)/5 - ...
(and we can eliminate C now by plugging in 0 for x and getting ln(1) = C = 0)

now, using some substitution, take our new function g(x) = ln(1+x) and its taylor series, and substitute x² for x. that is, g(x²) = ln(1 + x²), and its taylor series becomes (x²) - ((x²)²)/2 + ((x²)³)/3 - ((x²)⁴)/4 + ((x²)⁵)/5 - ... (placing x²everywhere that we see an x), which equals
x² - (x⁴)/2 + (x⁶)/3 - (x⁸)/4 + ...

thus ln( 1 + x² ) = x² - (x⁴)/2 + (x^6)/3 - (x⁸)/4 + (x^10)/5 - ...
and in general form:
∑ (-1)ⁿ⁻¹ * x²ⁿ/n

[from n=1 to ∞]

though this method seems very confusing, it will get easier and more natural with practice. the trick to picking the correct initial function will also come with practice and your familiarity with taylor series.

i hope this wasn't too confusing!

Sunspots, differential rotation, the dynamo effect, and the Babcock Model

This is an explanation of a few phenomena that I gave to my friend to help her for her astronomy class. the main topics are those in the title of this post

(I used mainly wikipedia as a source for the specifics, but I found that my own knowledge of physics became VERY useful. and i was also very surprised to learn that this is being taught as an introductory astronomy class, which required no background in physics! how odd...):

okay, well all of these things are obviously related, so i'll try to explain them that way.

first, i’ll briefly give you a good way to think about convection currents. you deal with convection currents pretty much every day (or at least once a week, probably). when you boil a pot of cold water, there is a heat source at the bottom (fire) heating up the water. the water at the bottom of the pot then becomes hot, and since hot water (like hot air) rises (most reasonably because it is less dense), the hot water rises to the surface of the pot. this means that the cold water on top is forced down to bottom of the pot. this water then gets heated while the hot water on top is cooling off. and again, the hot water at the bottom rises, and the cold water at the top sinks down. as you can see, this process repeats itself over and over, creating a circular kind of flow, or current, of water from the bottom of the pot to the top and back down again, caused by thermal convection. therefore, we call this movement a 'convection current'. viola! now, there are many types of convection, but this is the general principle behind all of it. convection currents in the sun are caused mainly by thermal convection (seeing as there are hundreds of very very hot nuclear exposions going on in there all the time), and probably some other types as well.

these convection currents in the sun are what are responsible for differential rotation. differential rotation means that one part of the sun, or the sun's surface, is rotating faster than another.

to picture this, imagine a bunch of strings all laid out straight from one point - kind of like a spider web (or if you drew a bunch of rays coming out from a point). now if you start to rotate the thing, but keeping everything rotating at the same speed, it will look the same all the time because all of the strings remain spread out straight.

but now if you rotate it, say, faster in the middle and slower at the ends, you'll see the structure start to warp and bundle, yes? since the inside of the web is spinning faster (making more rotations per second than the more outlying parts), the strings will look like they're making a bunch of arcs. [i've actually included a picture now, but i just made it in paint, so don't laugh]. this is essentially differential rotation. the speed of the rotation of part of the sun's surface is faster in some places than others, and this causes a warping (like our string).

interestingly, this is almost exactly what the Babcock Model describes. just imagine that these lines are "magnetic field lines". i know you don't have much physics, but let's just say for a second that these magnetic field lines cover the entire surface of the sun like our web does (imagine them as longitudinal lines). do you see how if we keep up this process of rotating one area faster than another, the lines become warped and eventually pretty tangled up? that's what is happening to the magnetic field lines on the surface of the sun. now, magnetic fields really don't like being tangled up like this, and it produces enormous magnetic fields and forces. these strong magnetic fields then have an effect on light emitting substances on the sun’s surface (plasmas, mostly), which restricts the emission of light in that area. therefore, these areas tend to appear different colors and darker. we call those areas sunspots.

so it's actually convection currents and differential rotation which explain the Babcock Model! cool!

so, i hope this has helped you understand a little more of what is going on in those first two things.

also, i just got all of this information from these two wikipedia sites (and whatever i know about physics already):

http://en.wikipedia.org/wiki/Babcock_Model

http://en.wikipedia.org/wiki/Dynamo_effect

...

well hmm, i've been thinking of a way to talk about the dynamo effect, but i can't get it quite right. can you tell me what you guys have learned about it? and what you need to know?

in the meantime, i'm gonna talk a little more about it.

from what i understand, the magnetic fields in the earth's dynamo are created by liquid metals (mostly iron), and in the sun it's ionized gas. that's about the only part that i see that has anything to do with gas. so i don't think gases have much to do with the geodynamo. is that correct?

so if we wanna know about the dynamo in the earth's case, we're gonna study metals, not gases (tell me if i'm wrong though, cause you might also need to know how the sun's dynamo works too - although it's pretty much the same thing).

so, we want to know why the earth's core can sustain (and produce) such strong magnetic fields for so long. (under 'normal' conditions it would be very difficult to maintain a dynamic magnetic field like the earth's for more than about 20,000 years - and contrary to the beliefs of creationists, strong evidence suggests that the earth is much, much older than that). the answer is that it is a magnetic dynamo. but what does that mean? well, it just so happens that the outer part of the earth's molten iron core is rotating. [the reason that it is spinning is a whole other topic in and of itself. but briefly, i can tell you that it is due to a lot of things, including heat currents, gravitational fluctuations, and radioactive materials]. so then, why does it matter if some liquid metal is rotating around on the outer part of the earth's core? well, iron is a very electrically conductive material - that is, electric charge and current can flow over and through it very easily.

so, this liquid iron core (remember 'liquid' iron is just iron that is hot enough to melt) has a bunch of electric charge running up and down and over it. and, it is a basic principle of electromagnetism that when such charge is spinning (much like the molten iron core is spinning), the charge creates a magnetic field! furthermore, as long as the core (and the electric charge) is spinning, it will continue to produce a magnetic field. so this explains how the earth has been able to maintain a magnetic field for so long!

and there's a lot of theory about how and under what conditions the core needs to be rotating to produce this effect, but that is all very complicated and i don't think you need to know it.

so, while i hope this was useful, i know it's all very abstract physics, so i hope is was able to make some of it concrete for you

yeah, so tell me if i'm even talking about the stuff you need to know here! and ask anything else or clarification if you need it!

:)

again, a lot of this information is from wikipedia, including two more articles that i found which might be useful, but they're less relevant:

http://en.wikipedia.org/wiki/Tachocline

http://en.wikipedia.org/wiki/Convection

NOTE:

unfortunately, I do not yet know how to upload pictures to my blog, so you guys will have to miss out on my amazing mspaint skills for now

Hello, world!

Hi everybody!

My name is Nick; this is a brief introduction to who i am and what this blog is about (also a test to see how this blog thing works).

I am a first year college student and an aspiring physicist

Needless to say, i like math, physics, chemistry, programming, computers, and pretty much all sciencey-related stuff (and sometimes even grammar).

Well, i guess i'll see how this turns out.

talk to you later, internet