if N < 40559.850189135206

1. Started with
   \[\log \left(N + 1\right) - \log N\]
   0.1

if 40559.850189135206 < N

1. Started with
   \[\log \left(N + 1\right) - \log N\]
   59.5
2. Applied taylor to get
   \[\log \left(N + 1\right) - \log N \leadsto \left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^2}\]
   0.0
3. Taylor expanded around inf to get
   \[\color{red}{\left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^2}} \leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^2}}\]
   0.0
4. Applied simplify to get
   \[\color{red}{\left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^2}} \leadsto \color{blue}{\frac{1}{N} + \left(\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} - \frac{\frac{1}{2}}{N \cdot N}\right)}\]
   0.0
5. Applied simplify to get
   \[\frac{1}{N} + \color{red}{\left(\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} - \frac{\frac{1}{2}}{N \cdot N}\right)} \leadsto \frac{1}{N} + \color{blue}{\left(\frac{\frac{1}{3}}{{N}^3} - \frac{\frac{1}{2}}{N \cdot N}\right)}\]
   0.0