How do the partial sums of the alternating harmonic series behave relative to the limit $\ln (2)$?

Conditions

• The series is the alternating harmonic series $\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n}$.
• The limit of the series is $\ln(2)$.

Reasoning, step by step

1. Plot the sequence of partial sums against the number of terms $n$.
2. Observe that the points jump above and below the horizontal line $y = \ln(2)$.
3. Note that the distance between the partial sums and $\ln(2)$ decreases as $n$ increases.
4. Conclude that the convergence is oscillatory, approaching the limit from alternating sides.

Example

The animation traces the path of the partial sums as yellow dots connected by lines, jumping above and below the dashed line labeled $\ln(2)$.

Common misconceptions

• Believing that partial sums approach the limit monotonically.
• Thinking that the series diverges because the partial sums oscillate.

Watch the explanation

Connected concepts

Explore next

• Why do odd and even partial sums approach the limit from opposite sides?
• How does the error bound relate to this oscillatory behavior?