# Four Examples

Standard

Following are the four examples of sequences (along with their properties) which can be helpful to gain a better understanding of theorems about sequences (real analysis):

• $\langle n\rangle_{n=1}^{\infty}$ : unbounded, strictly increasing, diverging
• $\langle \frac{1}{n}\rangle_{n=1}^{\infty}$ : bounded, strictly decreasing, converging
• $\langle \frac{n}{1+n}\rangle_{n=1}^{\infty}$ : bounded, strictly increasing, converging
• $\langle (-1)^{n+1}\rangle_{n=1}^{\infty}$ : bounded, not converging (oscillating)

I was really amazed to found that $x_n=\frac{n}{n+1}$ is a strictly increasing sequence, and in general, the function $f(x)=\frac{x}{1+x}$ defined for all positive real numbers is an increasing function bounded by 1:

The graph of x/(1+x) for x>0, plotted using SageMath 7.5.1

Also, just a passing remark, since $\log(x)< x+1$ for all $x>0$, and as seen in prime number theorem we get an unbounded increasing function $\frac{x}{\log(x)}$ for $x>1$

The plot of x/log(x) for x>2. The dashed line is y=x for the comparison of growth rate. Plotted using SageMath 7.5.1

# Prime Polynomial Theorem

Standard

I just wanted to point towards a nice theorem, analogous to the Prime Number Theorem, which is not talked about much:

# irreducible monic polynomials with coefficients in $\mathbb{F}_q$ and of degree $n \sim \frac{q^n}{n}$, for a prime power $q$.

The proof of this theorem follows from Gauss’ formula:

# monic irreducible polynomialswith coefficients in $\mathbb{F}_q$ and of degree $n$ = $\displaystyle{\frac{1}{n}\sum_{d|n}\mu\left(\frac{n}{d}\right)q^d}$, by taking $d=n$.

For details, see first section of this: http://alpha.math.uga.edu/~pollack/thesis/thesis-final.pdf