sum formula for 1/x^2 + 1/x^2 + 1/x^3 + ...?

[deemanw]

what is formula for 1/x^1 + 1/x^2 +1/x^3 + .......................

 

[soroban]

Re: last question - sum formula?

Hello, deemanw!

What is formula for: \(\displaystyle \L\,\frac{1}{x^1} \,+ \,\frac{1}{x^2}\,+\,\frac{1}{x^3}\,+\,\frac{1}{x^4}\,+\,\cdots\)

The sum of an infinite geometric series is: \(\displaystyle \L\,S \;=\;\frac{a}{1 - r}\)
$\displaystyle \;\;$where $\displaystyle a$ is the first term and $\displaystyle r$ is the common ratio.

We are given a geometric series with: $\displaystyle \,a\,=\,\frac{1}{x}$ and $\displaystyle r\,=\,\frac{1}{x}$

Therefore, the sum is: \(\displaystyle \L\,S\;= \;\frac{\frac{1}{x}}{1\,-\frac{1}{x}} \;= \;\frac{1}{x\,-\,1}\)

 

[deemanw]

Thank you for helping me!