Series Math Example 1

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

Example 1

easy
Compute partial sums S1S_1 through S4S_4 for ∑n=1∞12n\sum_{n=1}^{\infty} \frac{1}{2^n} and identify the limit.

Solution

  1. 1
    Terms: 12,14,18,116,…\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots
  2. 2
    S1=12S_1 = \frac{1}{2}, S2=34S_2 = \frac{3}{4}, S3=78S_3 = \frac{7}{8}, S4=1516S_4 = \frac{15}{16}.
  3. 3
    Pattern: Sn=1−12n→1S_n = 1 - \frac{1}{2^n} \to 1.
  4. 4
    Alternatively, geometric series: a=12a = \frac{1}{2}, r=12r = \frac{1}{2}, sum =a1−r=1= \frac{a}{1-r} = 1.

Answer

S1=12S_1=\frac{1}{2}, S2=34S_2=\frac{3}{4}, S3=78S_3=\frac{7}{8}, S4=1516S_4=\frac{15}{16}; series sum =1= 1
The partial sums approach 1, confirming the series converges. Each new term adds half the remaining gap to 1.

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 2 hard

Show that the harmonic series [formula] diverges.

Example 3 easy

Write the first four partial sums of [formula]

Example 4 medium

Use the divergence test on [formula].

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