Intereting Posts

Can a field be isomorphic to its subfield but not to a subfield in between?
Deducing PA's axioms in ZFC
Sum of Stirling numbers of both kinds
Intuition for Smooth Manifolds
Physical interpretation of Laplace transforms
nilpotent subgroup and hypercenter
Ideal of cusp forms for $\Gamma_0(4)$ is principal
Lost in terminology: What is the meaning of the words “Constraint” and “Parameter” in a goodness of fit?
What does “isomorphic” mean in linear algebra?
The fundamental group of a topological group is abelian
Generalization of the Jordan form for infinite matrices
a formula involving order of Dirichlet characters, $\mu(n)$ and $\varphi(n)$
The formula for a distance between two point on Riemannian manifold
Proof that plane with $N$ lines can be painted with two colors so that any two neighboring regions are painted in different colors
Solving Some Transcendental Equations

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:

$$\sum_{n=1}^{\infty}\frac{3}{n(n+3)}$$

- How find this equation $\prod\left(x+\frac{1}{2x}-1\right)=\prod\left(1-\frac{zx}{y}\right)$
- Calculus 2 integral $\int {\frac{2}{x\sqrt{x+1}}}\, dx$
- Find the radius of convergence of $\sum_{n=1}^{\infty}{n!x^{n!}}$
- Summation using residues
- How do “Dummy Variables” work?
- How to prove that $\log(x)<x$ when $x>1$?

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 $n$ terms and I find myself lost.

Here is the what I’m talking about:

$$S_n=\sum_{i=1}^{n}\frac{3}{i(i+3)}=\sum_{i=1}^{n}\left(\frac{1}{i}-\frac{1}{i+3} \right)$$

The next few terms are shown to be this:

$$=\left(1-\frac{1}{4}\right)+\left(\frac{1}{2}-\frac{1}{5}\right)+\left(\frac{1}{3}-\frac{1}{6}\right)+\left(\frac{1}{4}-\frac{1}{7}\right)+..+$$

And it continues but this is the part where I get confused…

$$\left(\frac{1}{n-3}-\frac{1}{n}\right)+\left(\frac{1}{n-2}-\frac{1}{n+1}\right)+\left(\frac{1}{n-1}-\frac{1}{n+2}\right)+\left(\frac{1}{n}-\frac{1}{n+3}\right)$$

Where did the $n$ terms in the denominator come from?

- Differentiable Strictly Convex Function on Interval
- Is there a general formula for the derivative of $\exp(A(x))$ when $A(x)$ is a matrix?
- $\frac{1}{\infty}$ - is this equal $0$?
- How does partial fraction decomposition avoid division by zero?
- Find the domain of $x^{2/3}$
- Generalized binomial theorem
- Calculate: $\lim\limits_{x \to \infty}\left(\frac{x^2+2x+3}{x^2+x+1} \right)^x$
- Solving $-u''(x) = \delta(x)$
- Proof of $\int_0^\infty \frac{x^{\alpha}dx}{1+2x\cos\beta +x^{2}}=\frac{\pi\sin (\alpha\beta)}{\sin (\alpha\pi)\sin \beta }$
- Closed-form of $\int_0^1 \left(\ln \Gamma(x)\right)^3\,dx$

$$\sum_{n=1}^{\infty}\frac{3}{n(n+3)}=\lim_{k\to\infty}\sum_{n=1}^{k}\frac{3}{n(n+3)}=$$

$$=\lim_{k\to\infty}\left(\frac{11}{6}-\left(\frac{1}{k+1}+\frac{1}{k+2}+\frac{1}{k+3}\right)\right)=\frac{11}{6}$$

because

$$\sum_{n=1}^{k}\frac{3}{n(n+3)}=\sum_{n=1}^{k}\left(\frac{1}{n}-\frac{1}{n+3}\right)=\sum_{n=1}^{k}\frac{1}{n}-\sum_{n=1}^{k}\frac{1}{n+3}=$$

$$=1+\frac{1}{2}+\frac{1}{3}+\sum_{n=4}^{k}\frac{1}{n}-\sum_{n=1}^{k}\frac{1}{n+3}=\frac{11}{6}+\sum_{n=4}^{k}\frac{1}{n}-\sum_{j=4}^{k+3}\frac{1}{j}=$$

$$=\frac{11}{6}+\sum_{n=4}^{k}\frac{1}{n}-\left(\sum_{j=4}^{k}\frac{1}{j}+\frac{1}{k+1}+\frac{1}{k+2}+\frac{1}{k+3}\right)=$$

$$=\frac{11}{6}-\left(\frac{1}{k+1}+\frac{1}{k+2}+\frac{1}{k+3}\right)$$

- Prove by induction that $n^5-5n^3+4n$ is divisible by 120 for all n starting from 3
- Proof of an elliptic equation.
- Travelling salesman problem as an integer linear program
- How to simplify $\sum_{a_1=1}^n\sum_{a_2=1}^{a_1}\sum_{a_3=1}^{a_2}\dots\sum_{a_{k+1}=1}^{a_k}1$
- The nil-radical is an intersection of all prime ideals proof
- Bijection between derangements and good permutations
- How to explain for my daughter that $\frac {2}{3}$ is greater than $\frac {3}{5}$?
- Express roots in polynomials of equation $x^3+x^2-2x-1=0$
- In $ \triangle ABC$ show that $ 1 \lt \cos A + \cos B + \cos C \le \frac 32$
- Examine convergence of $\sum_{n=1}^{\infty}(\sqrt{a} – \frac{\sqrt{b}+\sqrt{c}}{2})$
- Connected metric spaces with at least 2 points are uncountable.
- Prime dividing the binomial coefficients
- Why do we need noetherianness (or something like it) for Serre's criterion for affineness?
- What is the Coxeter diagram for?
- How do I teach university level mathematics to myself?