Estimating error using comparison/limit comparison test

[motornoob101]

I am confused about the right formula for this. Is it

R_{n} \leq T_{n} \leq \int_{n+1}^{\infty} or R_{n} \leq T_{n} \leq\int_{n}^{\infty}?

Say for example, I want to estimate the error in using the sum of the first 10 terms to approximate the sum of the series.

The textbook seems to use both methods (Use n for the p-series and n+1 for the geometric series)

(as seen in the examples here)

http://p3t3rl1.googlepages.com/textsolutionconfused.jpg

and I am quite confused which it is the right way. I am wondering if someone could clear this up for me. Thanks!

 

[fresh_42]

It is always best to draw a picture, not an exact one, but one that contains the crucial rectangles. The index depends on how the integration area is split into equidistant parts and whether you start counting by $0$ or by $1$.