Infinite series estimation using integral test

[motornoob101]

Homework Statement

Estimate $$\sum^{\infty}_{n=1}n^{-3/2}$$ to within 0.01

Homework Equations

$$\int^{\infty}_{n+1}f(x)dx\leq R_{n} \leq \int^{\infty}_{n}f(x)dx$$

The Attempt at a Solution

So my strategy was using the above formula to find Rn, where Rn = 0.01 or 1/10^2. Then that will give me the n value, which I can use to find the partial sum. It worked for all other problems but when I looked at the solution manual, they are doing something weird and I can't understand.

$$\int^{\infty}_{x}x^{-3/2}dx= \left[ -2x^{-1/2}\right]^{\infty}_{x}$$

and if I do the appropriate substitution etc.. I get $$\frac{2}{\sqrt{x}} = \frac{1}{10^2}$$, which give me a x or n value of 40000. A bit too big considering the textbook has a answer of like 14. What I am doing wrong? Thanks.

Here is the textbook's solution, which I don't get at all..

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

 

[HallsofIvy]

The crucial part in understanding "what they are doing" is where the say "From the end of example 6, we see that the error is at most half of the interval". What do they say at "the end of example 6"?

 

[motornoob101]

Ok, I don't know what they did in example 6. They had just one sentence about the error and it is very unclear. What I am curious though, and which is the whole point of my question to begin with, is what I am doing correct? Thanks.

 

[rootX]

I haven't gone this far into series,
but I think your limits are wrong.
You should go from 1 to t, where t --> inf
(We don't just put inf on top and evaluate)