Converging to 5: Solving a Tricky Calculus III Series Problem

[rman144]

I've been working on this for two hours and have had zero luck:

Given:

sum{k=1 to k=oo} [((-1)^(k+1))/k]

Rearrange the terms so the series converges to 5 [lol, I haven't a clue how].

 

[Mark44]

Take a look at this Wikipedia article: http://en.wikipedia.org/wiki/Riemann_series_theorem
The reason you can use this theorem is that your series is conditionally convergent but not absolutely convergent.

BTW, here is your series using LaTeX code:
\sum_{k = 1}^\infty \frac{(-1)^{k + 1}}{k}

 

[HallsofIvy]

Separate even (positive) terms as a_n and odd (negative) terms, as b_n Then your series itself is a_n+ b_n while the absolute value is a_n- b_n. You can show the the series involving a_n only goes to infinity while the series involving only b_n goes to negative infinity. Okay, take series only from a_n until the sum is greater than 5. Since that sum minus 5 is a finite number, you add take terms from b_n until that sum is back less than 5. Now add terms from a_n until it is back larger than 5, etc.