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Q:

Taylor series having an odd power

Can someone please explain, why this power is odd?
$\sum_{n=1}^{\infty}\frac{(-1)^n\sin(nx)}{n^3} =
\sum_{n=0}^{\infty}\frac{(-1)^{n+1}}{n!}\frac{d^n}{dx^n}(2x-x\sin x)
=\frac{2}{3!}\frac{d^3}{dx^3}x(2x-x\sin x)+\frac{1}{2!}\frac{d^2}{dx^2}(x\cos x)$

A:

It’s pretty easy to see that if you differentiate the general form of a product of $f(x)$ and $g(x)$ again, you will get terms of the form
$$
\left(\frac{d^n}{dx^n}f(x)\right)\left(\frac{d^m}{dx^m}g(x)\right)
$$
for any $n,m\in\mathbb{N}$.
Now, in your case, you’re dealing with a product of $\sin x$ and another function. A straightforward calculation gives that the product of two sine functions is a function of the form $2\cos x\sin x -\sin 2x$. Differentiating this three times gives
$$
6\cos x\sin x – 4\cos x\cos 2x-2\cos 2x\sin x-\sin 3x
$$
which has the same general form as the above product; the only difference is that now $n$ is not restricted to be even.
But that’s not all. The way you’ve written your equality is slightly ambiguous: you’re differentiating on the LHS with respect to $x$ of the given function, but, on the RHS, you’re differentiating the coefficients with respect to $x$. (This is a very minor problem, and the two expressions are equivalent.)
Now if you differentiate the coefficients with respect to $x$, you get
$$
\left(\frac{d^3}{dx^3}\sin x\right)\left(-\cos x-\sin x\cos x\right)+
\left(\frac

Distribution of polychlorinated biphenyl congeners in fish species from the Lower St. Lawrence River, eastern Canada.
Chlorinated hydrocarbons, such as polychlorinated biphenyls (PCBs), bioaccumulate through the food web and pose a risk to human health. In this study, we determined the occurrence and distribution of the most common PCB congeners in 11 fish species from the Lower St. Lawrence River, eastern Canada. Only PCB congener 118 was detected in all species tested (100%) and at all sites (96%). PCB congeners 101, 138, and 180 were detected at most sites. Congeners 153 and 180 were not detected, whereas congeners 138 and 170 were detected in the highest percentages (86% and 65%, respectively). As the concentrations of the other tested congeners were at or below the detection limit, it is likely that more congeners were present at some sites. This difference could be explained by the point and time of collection, as well as by the difference in the ability of congeners to bioaccumulate through trophic levels. In fact, bioaccumulation rate was species specific, with trout being significantly greater than lumpfish and Atlantic salmon. The trophic level (TL) of a fish species determines the relative bioaccumulation rate of its diet. Most of the PCBs detected in these species were characterized by a low to medium bioaccumulation. The main factors affecting bioaccumulation rate are the TL and the bioconcentration factor (BCF) of the species. Thus, the greater bioaccumulation in trout can be explained by the higher BCF (19 vs. 1.4) and the higher TL (3-5 vs. 1.1).The present invention relates to a method and system for predicting the flammability of flammable fluids. More particularly, the invention relates to a method and system for predicting the flammability of flammable fluids that are useful for detecting leaky underground storage tanks.
Underground storage tanks are used to contain hazardous liquids. They are used in industries where certain materials must be stored under a certain pressure or temperature. These materials may be oil, gasoline, propane, and wastewater. These underground storage tanks are identified by a number of different types of leak detection sensors. Different types of leak detection sensors are used in different underground storage tanks. It is important to be able to predict the flammability of these flammable fluids to avoid

Windows XP:
Windows Vista:
8 GB free HDD space
DirectX 9.0c graphics card, Windows 2000 compatibility not guaranteed
Adobe Flash Player 9.0 (Mac users must use version 10.0.3)
Wine 1.2.0 or above
System requirements are subject to change. All the code in this demo is LGPLv3.
For best results: Windows XP (32 bit) or Windows Vista (32 bit)
DirectX 9.0c graphics card
(Mac

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