Proving pi/4-ln(sqrt(2)) with Arctan Series

[EV33]

Homework Statement

Using the power series for arctan show that pi/4-ln(sqrt(2))=1-1/2-1/3+1/4+1/5...

Homework Equations

arctan=1-(1/3)x^3+(1/5)x^5...(((-1)^n)(x^(2n+1)))/(2n+1)

The Attempt at a Solution

The first thing I noticed was that arctan(1)=pi/4 and represented 1-1/3+1/5...

I am having trouble figuring out what -1/2+1/4-1/6... is equivalent to besides a simple series.

If anyone has any hints. I would love to hear them because I am stuck.

 

[EV33]

I have tried converting it back to a geometric series and comparing it to the natural log taylor series but haven't had any luck.

 

[Char. Limit]

What's the sum from n=1 to infinity of $$\frac{\left(-1\right)^n}{n}$$? Also, do you see that $$log(\sqrt{2}) = \frac{1}{2}log(2)$$?

 

[EV33]

Thank you so much. It is always something simple. That log property makes this problem a lot easier.