Alternating series test for convergence

[Maddie1609]

Homework Statement

Homework Equations

The Attempt at a Solution

I don't get how they got what's stated in the above picture. Where does 1/2 and n/(n + 1) come from? Can't you just show that a_{n + 1} ≤ a_n?

 

[mfb]

Where does 1/2 and n/(n + 1) come from? Can't you just show that a_{n + 1} ≤ a_n?

That's what they do. The logic is easier to follow if you start with the result and reduce this to something true (look at the steps in reverse order), but this direction works as well. The first line is then a clever guess what will be needed later (and the statement is clearly true).

 

[Maddie1609]

That's what they do. The logic is easier to follow if you start with the result and reduce this to something true (look at the steps in reverse order), but this direction works as well. The first line is then a clever guess what will be needed later (and the statement is clearly true).

A lot easier to follow that direction, thanks! The final step has 2ⁿ and 2^{n - 1} instead of (-2)ⁿ and (-2)^{n - 1} which I don't get.

 

[mfb]

The test compares the magnitude of the sequence elements only, powers of -1 don't change the magnitude.
The sign has to be checked separately (easy here).

 

[Maddie1609]

The test compares the magnitude of the sequence elements only, powers of -1 don't change the magnitude.
The sign has to be checked separately (easy here).

Oh okay! Thank you