Series convergence

According to the text annotations in the video, the nth harmonic number $H_n$ can be approximated by the natural logarithm of $n$ plus Euler-Mascheroni constant $\gamma$. The specific formula provided is $H_n \approx \ln(n) + \gamma$.

Conditions: Analyzing the limit behavior of $H_n$; Large $n$

The series converges because its partial sums $S_n$ approach a specific finite value, which is 2. As shown in the video, the formula for the partial sum is $S_n = 2(1 - \frac{1}{2^{n+1}})$.

Conditions: First term $a=1$; Common ratio $r=1/2$

To calculate $S_n$, use the formula $S_n = 2(1 - 2^{-(n+1)})$. Note that the summation starts from $k=0$ to $n$, which means there are $n+1$ terms in total.

Conditions: Summation index $k$ runs from 0 to $n$; Ratio $r = 1/2$

The harmonic series diverges because its partial sums continue to rise indefinitely without bound. Although the individual terms $\frac{1}{k}$ become smaller and approach zero, they do not decrease fast enough to keep the total sum finite.

Conditions: Series is the harmonic series $H_n = \sum_{k=1}^n \frac{1}{k}$; Individual terms approach 0 as $k \to \infty$