What is the asymptotic approximation for the nth harmonic number $H_n$ according to the video's formal analysis?

Conditions

• Analyzing the limit behavior of $H_n$
• Large $n$

Reasoning, step by step

1. Refer to the formal text annotations appearing at the end of the video.
2. Identify the expression associated with the divergent case.
3. Extract the approximation formula involving natural log and gamma.
4. Interpret the result as indicating slow, unbounded growth.

Example

The script states: 'They show the limit expressions for both sequences and provide explicit formulas... and the approximation $H_n \approx \ln(n) + \gamma$ for the divergent one.'

Common misconceptions

• Confusing $\ln(n)$ with $\log_{10}(n)$.
• Believing the approximation becomes exact for any finite $n$.

Watch the explanation

Connected concepts

Explore next

• What is the numerical value of the constant $\gamma$?
• How does this approximation help in estimating how large $n$ needs to be for $H_n > 10$?