- #1
Berrius
- 19
- 0
Homework Statement
As part of a problem I have to show that [tex]lim_{n\to\infty}\sum_{i=\frac{n}{2}}^{n}\frac{1}{i}=ln(2)[/tex]
Homework Equations
Taylor expansion of ln(2): [tex]\sum_{i=1}^{\infty}\frac{(-1)^{k+1}}{k}[/tex]
The Attempt at a Solution
ln(2) can be written as: [tex]ln(2) = \sum_{i=1}^{\frac{n}{2}}\frac{(-1)^{k+1}}{k} + \sum_{i=\frac{n}{2}+1}^{n}\frac{(-1)^{k+1}}{k} + \sum_{i=n+1}^{\infty}\frac{(-1)^{k+1}}{k}[/tex]
Where the middle term looks a lot like the sum i need. However I need some way to get rid of the alternating sign change, but i don't see how.