Articles of partial fractions

Converting multiplying fractions to sum of fractions

I have the next fraction: $$\frac{1}{x^3-1}.$$ I want to convert it to sum of fractions (meaning $1/(a+b)$). So I changed it to: $$\frac{1}{(x-1)(x^2+x+1)}.$$ but now I dont know the next step. Any idea? Thanks.

How to properly set up partial fractions for repeated denominator factors

This question already has an answer here: Derivation of the general forms of partial fractions 7 answers

Understanding the solution of a telescoping sum $\sum_{n=1}^{\infty}\frac{3}{n(n+3)}$

I’m having trouble understanding infinite sequence and series as it relates to calculus, but I think I’m getting there. For the below problem: $$\sum_{n=1}^{\infty}\frac{3}{n(n+3)}$$ The solution shows them breaking this up into a sum of partial fractions. I understand how they got the first two terms, but then they show the partial fractions of the […]

Partial fraction expansion of $\frac{1}{x(x+1)(x+2)\cdots(x+n)}$

I try to find a partial fraction expansion of $\dfrac{1}{\prod_{k=0}^n (x+k)}$ (to calculate its integral). After checking some values of $n$, I noticed that it seems to be true that $\dfrac{n!}{\prod_{k=0}^n (x+k)}=\sum_{k=0}^n\dfrac{(-1)^k{n \choose k}}{x+k}$. However, I can’t think of a way to prove it. Can somebody please help me?

Integral of rational function with higher degree in numerator

How do I integrate this fraction: $$\int\frac{x^3+2x^2+x-7}{x^2+x-2} dx$$ I did try the partial fraction decomposition: $$\frac{x^3+2x^2+x-7}{x^2+x-2} = \frac{x^3+2x^2+x-7}{(x-1)(x+2)}$$ And: $$\frac{A}{(x-1)}+\frac{B}{(x+2)}=\frac{A(x+2)}{(x-1)(x+2)}+\frac{B(x-1)}{(x-1)(x+2)}$$ Then: $$\ A(x+2) + B(x-1)= x^3+2x^2+x-7$$ When I do this I gets a wrong answer, so I figured out that this may only work when the degree of the denominator is greater than the degree […]

Partial fraction decomposition of $ \frac{\pi}{\sin \pi z}$

I want to prove the partial fraction decomposition: \begin{align}\frac{\pi}{\sin \pi z} = \frac{1}{z} + 2z\sum\limits _{n=1} ^{\infty} \frac{(-1)^n}{z^2-n^2} \end{align} with the help of the partial fraction decomposition of \begin{align} \pi \cot\pi z = \frac{1}{z} + 2z\sum\limits _{n=1} ^{\infty} \frac{1}{z^2-n^2}. \end{align}

Decomposition into partial fractions of an inverse of a generic polynomial with three distinct roots.

Let $d \ge 2$ be an integer. Let $\left\{ m_j \right\}_{j=1}^d$ be strictly positive integers and $\left\{ b_j \right\}_{j=1}^d$ be parameters. Define the following quantity: \begin{equation} {\mathfrak F}_d(x) := \frac{1}{\prod\limits_{j=1}^d (x+b_j)^{m_j}} \end{equation} Below we decompose the quantity above into partial fractions for $d=3$ using differentiation with respect to the $b$-parameters. We have: \begin{eqnarray} &&{\mathfrak F}_d(x) […]

Find the sum $\sum_{n=1}^{50}\frac{1}{n^4+n^2+1}$

Find the sum $\sum_{n=1}^{50}\frac{1}{n^4+n^2+1}$ $$\begin{align}\frac{1}{n^4+n^2+1}& =\frac{1}{n^4+2n^2+1-n^2}\\ &=\frac{1}{(n^2+1)^2-n^2}\\ &=\frac{1}{(n^2+n+1)(n^2-n+1)}\\ &=\frac{1-n}{2(n^2-n+1)}+\frac{1+n}{2(n^2+n+1)}\\ \end{align}$$ For $n={1,2,3}$ it is not giving telescooping series. $=\frac{1}{3}+\frac{3}{14}+\frac{2}{13}+0-\frac{1}{6}-\frac{1}{7}$

Calculus Integral from Partial Fractions

When you have an irreducible quadratic factor repeated you can get integrals that look like $\int \dfrac{dx}{(x^2+a)^m}$, where $m>1$, integer, and $a>0$. What is the best way to integrate this function? Is there more than one way?

How did Euler prove the partial fraction expansion of the cotangent function: $\pi\cot(\pi z)=\frac1z+\sum_{k=1}^\infty(\frac1{z-k}+\frac1{z+k})$?

As far as we know, Euler was the first to prove $$ \pi \cot(\pi z) = \frac{1}{z} + \sum_{k=1}^\infty \left( \frac{1}{z-k} + \frac{1}{z+k} \right).$$ I’ve seen several modern proofs of it and they all seem to rely either on the Herglotz trick or on the residue theorem. I recon Euler had neither nor at his […]