what makes a Harmonic Series "harmonic?" (Calculus)?
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http://mathworld.wolfram.com/HarmonicSeries.html
681952
Here you go:\nhttp://en.wikipedia.org/wiki/Fibonacci_series
437819
Actually, you get X''/X=C and Y''/Y=-C where C does not have to be 0. Also, remember that sums of solutions are solutions, so the solutions you get from separation of variables only give a basis for the vector space of solutions, not all the solutions. The specifics are difficult in this forum, but you have to set up the series and use the boundary conditions to find the coefficients of the series.
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The function f(x)=Sigma of((f^n)(c)/n!)*((x-c)^n) is what you start with.  You can replace the Sigma(sum) with an integral.  Derivatives are the basically the opposite of Integrals.  If you derive the integral, you end up back at the start: ((f^n)(c)/n!)*((x-c)^n).  It might be easier to use a graph to see what happens. \n\nBut basically, intervals are merely used to look at a part of whatever is on the graph.  It doesn't matter what you do to that part, you are still looking at that part.  That's why the intervals don't change.\n\nDoes that help you or confuse you?
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Harmonic division is defined as "The division of a line segment by two points such that it is divided externally and internally in the same ratio". An illustration should help. From the ref., "Points C and D divide line segment AB harmonically if points C and D are on line AB and the following proportion is satisfied: AC/CB = AD/DB. If points C and D divide segment AB harmonically, it follows that points A and B divide segment CD harmonically."\nAs an example, mark points A, C, B, D in that order on a horizontal line segment. \n__A______C__B_____D__\nPutting A,C,B,D at, e.g., 0,3,4,6 satisfies the definition.
909919
A sinusoid is characterized by its amplitude and its period.  Your first graph has a variable amplitude, so you might want to take an average.
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The Fourier transform is a special case of the LaPlace transform.\n\nSpecifically s = jw, where w is the frequency in radians, j is square root of negative one (I'm an EE so we use j not i)
303213
Expressions such as yours is called a power tower. They are differentiable by writing them as exponentials but they are not integrable in closed form. You can find the series form at the source link however.
750000
The Riemann sum is a way of computing integrals by breaking down a curve into small area elements, approximating the area of each element, adding them together, then taking the limit as the number of elements goes to infinity.  You can find a good discussion with examples here http://en.wikipedia.org/wiki/Riemann_sum
510352
The solution by a_ebnlhaitham is not quite correct in that he omitted the 2 from the power of r (i.e.(teta)).The answer should be 1/(1 - r^2) not 1/(1 - r). The series converges because r lies in the open interval (-1,1).
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sin x = (1/2i)(e^ix-e^-ix)\ncos x = (1/2)(e^ix+e^-ix)\n\nsolutions of a simple harmonic oscilator are\ny = A sin x + B cos x or\ny = C e^ix + D e^-ix
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There is no function that exists that has as its derivative e^(x^2).  As someone else pointed out, you can use the Maclaurin series representation and integrate term by term.  However, this series converges very slowly and would require a lot of terms for two place decimal accuracy, especially if x is large.
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Simple harmonic motion or harmonic motion is a periodic vibration, for example, of a violin string or pendulum, that has a single frequency or an even multiple of one or is symmetrical about a point of equilibrium.
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Not sure if I understood your question. maybe you meant a series of functions. Suppose that, for each n you have a function f_n and, for every x in the domain of the functions f_n, you define S(x) = f_1(x) + ....f_n(x) +...provided this series converges. Then, you have a series of functions, since for every x S(x) is given by the limi of a series.\n\nFor example, consider s(x) = 1 + x  x^2/(2!) ....+ x^n(n!)....|This is a power series, in this case it converges for every real x. This is the definitioj of the exponencial function e^x
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The basic idea of a fourier series is to express a function - in this case the sawtooth wave - as a sum of sine and cosine functions of different frequencies (i.e. n omega0).  The result gotten for a sawtooth wave is the thing you described (the sideways 8 means "infinity").  That is, it's an infinite sum of sin curves, each one weighted by 1/n.  See a picture here: http://mathworld.wolfram.com/FourierSeriesSawtoothWave.html
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Do a Google search for "Fourier Transform."  I'm sure that someone has a proof on the web.
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I'm not familiar with that particular book, but the differential equations of simple harmonic motion are interesting.  You can make the system as complex as you like.  I've studied SHM in both maths and physics.  I simulated SH oscillators with operational amplifiers; in other words an analogue computer.  First you simulate a frictionless pendulum, then you introduce friction so the oscillations die away, then you introduce forcing terms of various frequencies to see what happens; like pushing a child on a swing.  If you're really clever, you can introduce hysteresis (delay), and extra degrees of freedom (flexible joints in the pendulum), and vary the Q of the system (how sharply you can tune it to the resonant frequency).  It's about the most open-ended project I ever saw.  I did it in second-year physics.  At its simplest, it's just sixth form maths, but you can turn it into an impossibly complex exercise in chaotic systems if you're clever.