Series Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Series.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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

Add up all the terms: a_1 + a_2 + a_3 + \ldots — an infinite series can still have a finite sum if terms shrink fast enough.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Infinite series can have finite sums (converge) or not (diverge).

Common stuck point: Terms going to zero isn't enough—the harmonic series 1 + \frac{1}{2} + \frac{1}{3} + \ldots diverges.

Sense of Study hint: Compute the first few partial sums S1, S2, S3, S4 and see whether they appear to stabilize or keep growing.

Worked Examples

Example 1

easy
Compute partial sums S_1 through S_4 for \sum_{n=1}^{\infty} \frac{1}{2^n} and identify the limit.

Solution

  1. 1
    Terms: \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots
  2. 2
    S_1 = \frac{1}{2}, S_2 = \frac{3}{4}, S_3 = \frac{7}{8}, S_4 = \frac{15}{16}.
  3. 3
    Pattern: S_n = 1 - \frac{1}{2^n} \to 1.
  4. 4
    Alternatively, geometric series: a = \frac{1}{2}, r = \frac{1}{2}, sum = \frac{a}{1-r} = 1.

Answer

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

Example 2

hard
Show that the harmonic series \sum_{n=1}^{\infty} \frac{1}{n} diverges.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Write the first four partial sums of 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots

Example 2

medium
Use the divergence test on \sum_{n=1}^{\infty} \frac{n}{2n+1}.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

sequence