- #1
arithmetic
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[itex]\sum_{m=0}^\infty \frac{(m-1)^{m-1}x^{m}}{m!}[/itex]
Interesting result...
Interesting result...
Last edited:
How about this? Integrate W wrt x:TheFool said:After looking at the graph, it's similar to the Lambert W function when |x|<1/e: [tex]-W(-x)=\sum_{m=1}^{\infty}{\frac{m^{m-1}x^{m}}{m!}}[/tex] Subtracting 1 from both sides will make it approximately equal to your sum. However, there is no way to manipulate my series to put yours in terms of the W function.
TheFool said:Well, it would seem I don't belong posting in this forum. I shouldn't have missed that.
arithmetic said:No, that`s wrong.
Yours is shorter and better. From what you stated, you just need one step further ...
and voilá, 1/ 1-...
Is this right?arithmetic said:[itex]\sum_{m=0}^\infty \frac{(m-1)^{m-1}x^{m}}{m!}[/itex]
micromass said:Did you actually know the answer to this problem??
A Taylor series is a mathematical representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.
Finding the function for a Taylor series allows us to approximate a complicated function with a simpler one that is easier to work with. It also helps us to understand the behavior of a function around a specific point.
To find the function for a Taylor series, we need to first determine the center point around which the series is based. Then, we calculate the derivatives of the function at that point and plug them into the general formula for a Taylor series.
The general formula for a Taylor series is f(x) = f(c) + f'(c)(x-c) + (f''(c)/2!)(x-c)^2 + (f'''(c)/3!)(x-c)^3 + ...
Taylor series are commonly used in mathematics, physics, and engineering to approximate functions, solve differential equations, and analyze the behavior of systems. They are also used in computer graphics and animation to create smooth curves and surfaces.