inequity involving geometric sum

[MathMad]

I have seen this from a book: [MATH]\sum_{n=1}^N\frac{1}{r^n}\leqslant\frac{r}{r-\delta}[/MATH] , where [MATH]\delta\lt[/MATH] r.
Now the RHS is equivalent to : [MATH]\frac{r}{r}\sum_{n=0}^\infty\left(\frac{\delta}{r}\right)^n[/MATH], so can I replace the [MATH]r[/MATH] in the LHS with [MATH]\delta\over r[/MATH]?
Since : [MATH]\frac{r}{r\left(1-{\delta\over r }\right)}=\sum_{n=0}^\infty\left(\frac{\delta}{r}\right)^n[/MATH]. We don't know whether either r or delta are greater or less than 1.

 

[Dr.Peterson]

Is your goal to prove the inequality, or to derive something else from it?

Do you mean replacing [MATH]r[/MATH] on both sides with [MATH]\delta\over r[/MATH]? And are you referring to the geometric series converging when [MATH]\delta\over r < 1[/MATH]? The latter is true if [MATH]\delta\lt[/MATH] r. (I'm assuming we also know that both numbers are positive.)

 

[MathMad]

Is your goal to prove the inequality, or to derive something else from it?

Do you mean replacing [MATH]r[/MATH] on both sides with [MATH]\delta\over r[/MATH]? And are you referring to the geometric series converging when [MATH]\delta\over r < 1[/MATH]? The latter is true if [MATH]\delta\lt[/MATH] r. (I'm assuming we also know that both numbers are positive.)

The aim is to explain and prove this. Ultimately it's part of a larger problem on analyticity of a function expressed as a Taylor series but that's not important. I could if it helps add picture of the page out of the book to give some context if you think it would help?

 

[MathMad]

2 pages From the book + cover. My problem is part of the inequalities on the same line at the top of page 94. I get the first on the LHS and the other sum involved.

 

[Dr.Peterson]

If the goal is to prove it, then you can't start with it.

But I think you have the right general idea; you want to derive it from the fact about geometric series. So start with the formula for a geometric series, and replace [MATH]r[/MATH] with [MATH]\frac{\delta}{r}[/MATH]. Then try to derive the stated inequality from what you get.

Give it a try, and show what you find.