Why is $1+1+\dfrac{1}{2^{\alpha-1}}+\dfrac{1}{2^{2(\alpha-1)}} +\cdots+\dfrac{1}{2^{(\alpha-1)(K-1)}}<1+\dfrac{1}{1-(1/2^{\alpha-1})}$?

Question

This is from Real Analysis, proof of Theorem 2.25:
"Let $\alpha$ be a real number. Then

$$\sum_{n=1}^{\infty}1/n^\alpha$$

is convergent if $\alpha>1$ and is divergent if $\alpha\le1$".
This was given information in the proof for the inequality of the series written on the topic $N\ge1$ ,$K\ge1$, and $1<\alpha<2$. I have put little effort understanding it but still couldn't understand why this is true and how did it just came out of a thin air.I hope experts like you can explain it and if you need any information related to this inequality please let me know in comment. Thanks in advance.

Answer 1

The sum of a finite GP with positive terms is always less than the corresponding infinite GP. See below: $$S=1+1+\frac{1}{2^{(a-1)}}+\frac{1}{2^{2(a-1)}}+\frac{1}{2^{3(a-1)}}+...+\frac{1}{2^{(a-1)(K-1)}}$$ $$=1+\sum_{j=0}^{K-1} 2^{-(a-1) j} < 1+\frac{1}{1-\frac{1}{2^{(a-1)}}}, a>1$$