As a follow up to a previous posting addressing the integral of $1/ (t^n+1)$ for $n\in \Bbb{N}$ I found the following
$$\int_0^ x \frac{1}{1+t^n}\, dt=\sum_{i=0}^{\infty}\frac{(i!)(n^i)x^{in+1}} {(x^n+1)^{i+1}\prod_{k=0}^i (kn+1)}$$
My son programmed the equation in CC++, and showed me that it works for $n=1,2$, 3... But as n gets large the computer cannot calculate.
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I would like assistance verifying if this equation is correct or wrong.
My request is also for others to demonstrate how the equation may be developed.
The relevance of the equation is that it generates multiple series/functions. Think of it as a function generator n. E.g.
At n= 1 the equation gives the series for $$LN(x+1)$$
At n= 2 the equation gives the series for $$ATAN(x)$$
The equation should be useful to other.
Where did you find that equation? It's quite different from what I got, which I shall explain now.
First, a common power series is
$$ \frac{1}{1-x} = \sum_{i\geq0} x^{i}. $$
Using the substitution $x=-t^{n}$,
$$ \frac{1}{1+t^{n}} = \sum_{i\geq0} (-t^{n})^{i} = \sum_{i\geq0} (-1)^{i}t^{in}. $$
Then,
$$ \int_{0}^{x} \frac{1}{1+t^{n}} dt = \int_{0}^{x} \sum_{i\geq0} (-1)^{i}t^{in} dt = \sum_{i\geq0} (-1)^{i} \int_{0}^{x} t^{in} dt. $$
The integral is simply
$$ \left[ \frac{t^{in+1}}{in+1} \right]_{0}^{x} = \frac{x^{in+1}}{in+1}. $$
Thus
$$ \int_{0}^{x} \frac{1}{1+t^{n}} dt = \sum_{i\geq0} (-1)^{i}\frac{x^{in+1}}{in+1}. $$
Any comments on where I may have gone wrong are appreciated.