Error estimate for alterning series

[danni7070]

Homework Statement

Find the smallest integer n that ensures that the partial sum $$s_n$$ approximates the sum s of the series with error less than 0.001 in absoulute value.

$$\sum_{n=1}^\infty (-1)^{(n+1)}\frac{n}{2^n}$$

Homework Equations

$${\abs{s-s_n}} <= s_{(n+1)} = a_{(n+1)}$$

(How can I do absolute value in LaTex?)

The Attempt at a Solution

I've got this far.

error <= first term = $$\frac{n+1}{2^{n+1}}$$

(because (-1)^n+1 doesn't matter in this example)

$$s - s_n < \frac{n+1}{2^n+1} < 0.001$$

Here I get lost.
From example in my textbook which says

$$a_n = \frac{1}{1+2^{n+1}}$$

This error is less than 0.001 if $$1+2^{(n+1)} > 0.001$$ Since $$2^{10} = 1024, n+1 = 10$$ will do. Hence n = 9.

In my example I have n on both sides of the fraction. So what do I do?

 

[Kreizhn]

Since n has to be an integer, I would say just brute force it.

 

[danni7070]

Brute force it? Do you mean just check in my calculator ? If you are referring to that, it doesn't help me since I'm not allowed to have anything in my test but a pencil and a paper.

 

[ZioX]

I wouldn't expect to be able to solve this analytically. All I can say is that I know it exists: (n+1)/2^{n+1} tends to zero monotonically.

 

[danni7070]

Well, I got the answer n = 9 by doing exactly as the example from the textbook but the answer is n = 13.

What does analytical and monotonical mean ?

 

[Kreizhn]

I got n = 13 via brute force.

Edit: In all fairness, it's not that hard to calculate the term when n = 13; the powers of 2 are pretty easy. I reckon any test question similar to this would have a more simple question in store.

 

[danni7070]

Yeah but you see Kreizhn I'm not allowed to use calculator so I have to use some method and solve the problem by hand. Sure, I can check if the answer is correct with my calculator but that doesn't help me when I get to the test where I'm not allowed to have anything but a pencil.

The problem for me is Algebra. I must have slept over or skipped some lectures few years ago when some Algebra rules were being tought.

I don't have any problem understanding the example from the book but I don't know how to find n in my problem. That's my question.

EDIT: It's true, the powers of 2 are really easy but when it comes to 6 or 8 or 112 what then? That is the reason I have to find some method finding n

 

[Kreizhn]

That is my point though; as ZioX had said, finding the solution to the previous problem is very difficult to do analytically even with a CAS. If you get a similar question on a test where you're not allowed to use a calculator, it's likely that the problem will be simplified such that you won't need one.