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.