Series Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy
Write the first four partial sums of 1−12+13−14+⋯1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots

Solution

  1. 1
    S1=1S_1 = 1.
  2. 2
    S2=12S_2 = \frac{1}{2}.
  3. 3
    S3=56S_3 = \frac{5}{6}.
  4. 4
    S4=712S_4 = \frac{7}{12}.

Answer

S1=1,  S2=12,  S3=56,  S4=712S_1=1,\; S_2=\frac{1}{2},\; S_3=\frac{5}{6},\; S_4=\frac{7}{12}
This alternating harmonic series converges to ln⁡2≈0.693\ln 2 \approx 0.693. Partial sums alternate above and below the limit, closing in from both sides.

About Series

The result of adding all the terms of a sequence together, either finitely or infinitely many terms.

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More Series Examples

Example 1 easy

Compute partial sums [formula] through [formula] for [formula] and identify the limit.

Example 2 hard

Show that the harmonic series [formula] diverges.

Example 4 medium

Use the divergence test on [formula].

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