Four Examples


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:


Thre graph of x/(1+x) for x>0. Plotted using SageMath 7.5.1

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


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