Intereting Posts

What are the important properties that categories are really abstracting?
Modus Operandi. Formulae for Maximum and Minimum of two numbers with a + b and $|a – b|$
Contraction of maximal ideals in polynomial rings over PIDs
Intuitive understanding of derangements.
Basis for the Riemann-Roch space $L(kP)$ on a curve
rational angles with sines expressible with radicals
locally compact metric space, regular borel measure
All linear combinations diagonalizable over $\mathbb{C}$ implies commuting.
How to solve a bivariate quadratic (not necessarily Pell-type) equation?
A non-abelian group such that $G/z(G)$ is abelian.
Find $\lim_{n \to \infty} \sqrt{n!}$.
Is $\sin^2(z) + \cos^2(z)=1$ still true for $z \in \Bbb{C}$?
Can a free group over a set be constructed this way (without equivalence classes of words)?
Toss a coin 10 times, run of 4 head occurs
Is there a simple method to prove that the square of the Sorgenfrey line is not normal?

Can I simplify or approximate this equation without sigma and combination?

\begin{align}

\sum_{i = 0}^n (-1)^i {n \choose i} \frac{{d+1}}{d(di + 1)}

\end{align}

- Is it possible to find $n-1$ consecutive composite integers
- Why is negative times negative = positive?
- Denesting Phi, Denesting Cube Roots
- How to deal with misapplying mathematical rules?
- Solving a literal equation containing fractions.
- Polynomial maximization: If $x^4+ax^3+3x^2+bx+1 \ge 0$, find the maximum value of $a^2+b^2$

- Where are good resources to study combinatorics?
- Proving a relation between $\sum\frac{1}{(2n-1)^2}$ and $\sum \frac{1}{n^2}$
- A new imaginary number? $x^c = -x$
- Prove that $\sum^{n}_{r=0}(-1)^r\cdot \large\frac{\binom{n}{r}}{\binom{r+3}{r}} = \frac{3!}{2(n+3)}$
- What is the number of compositions of the integer n such that no part is unique?
- Sum of reciprocals of binomial coefficients: $ \sum\limits_{k=0}^{n-1}\dfrac{1}{\binom{n}{k}(n-k)} $
- Show that $\, 0 \leq \left \lfloor{\frac{2a}{b}}\right \rfloor - 2 \left \lfloor{\frac{a}{b}}\right \rfloor \leq 1 $
- FoxTrot Bill Amend Problems
- Intriguing polynomials coming from a combinatorial physics problem
- How to factor the quadratic polynomial $2x^2-5xy-y^2$?

By the binomial theorem, $$(1-x^d)^n = \sum_{i=0}^n (-1)^i \binom{n}{i} x^{di}.$$

Now note that the integral $\int_{0}^1 x^{di}\,dx = \frac{1}{di+1}$. So

$$\int_0^1 (1-x^d)^n\,dx = \sum_{i=0}^n (-1)^i \binom{n}{i} \frac{1}{di+1},$$

which means that your sum is equal to $\frac{d+1}{d} \int_0^1 (1-x^d)^n\,dx.$

Finally, as pointed out by Jack D’Aurizio, we can express this in terms of the gamma function by making the substitution $z=x^d$, $dz = dx^{d-1} dx$ which changes the integral to

$$\frac{d+1}{d^2} \int_0^1 (1-z)^n z^{1/d – 1} dz = \frac{d+1}{d^2} B(n+1,\frac1d) = \frac{d+1\, \Gamma(n+1)\, \Gamma(\frac1d)}{d^2 \,\Gamma(n+1+\frac1d)} = \frac{\Gamma(n+1)\, \Gamma(2 + \frac1d)}{ \,\Gamma(n+1+\frac1d)}.$$

where $B$ is the classical beta function.

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