Over or Under? Analyzing the 50th Partial Sum of an Alternating Series

[fk378]

Homework Statement

Is the 50th partial sum, s_50, of the alternating series, "summation [(-1)^(n-1)] / n from 1-->infinity" an overestimate or an underestimate of the total sum? Explain

The Attempt at a Solution

First concern: Isn't every partial sum an underestimate for an increasing sequence and an overestimate for a decreasing sequence?

Secondly, I saw that b_n = 1/n, which is a divergent sum. So since it is increasing and divergent, wouldn't the partial sum s_50 be an underestimate?

This seems to be too easy a conclusion, so does anyone know if there is any other way to justify it?

 

[fk378]

The book says it is an underestimate.

 

[ToxicBug]

I think that its an overestimate, notice that its 1 - (1/2) + (1/3) - (1/4) + (1/5) - ... = 1 - (1/2 - 1/3) - (1/4 - 1/5) - ..., in reality you're always subtracting from 1. By the way, this sum is equal to ln(2).

Sorry for my previous post, I've messed up a bit :)