Series Formula

The Formula

S = \sum_{n=1}^{\infty} a_n = \lim_{N \to \infty} S_N \quad \text{where } S_N = \sum_{n=1}^{N} a_n

When to use: 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.

Quick Example

1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots = 2 (geometric series converges).

Notation

\sum a_n

What This Formula Means

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.

Formal View

Given a sequence (a_n), define partial sums S_N = \sum_{n=1}^{N} a_n. The series \sum_{n=1}^{\infty} a_n converges to S if \lim_{N \to \infty} S_N = S, i.e., \forall \epsilon > 0,\; \exists M : N > M \implies |S_N - S| < \epsilon.

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.

Common Mistakes

  • Concluding a series converges just because its terms approach zero: the harmonic series \sum \frac{1}{n} has terms going to 0 but still diverges to infinity.
  • Confusing the partial sum S_n = a_1 + a_2 + \cdots + a_n with the nth term a_n: the series converges if S_n approaches a limit, not just if a_n does.
  • Adding a finite number of terms and assuming the pattern of the partial sums will continue — the first 10 partial sums might appear to stabilize but the series can still diverge.

Why This Formula Matters

Foundation for advanced math: Taylor series, Fourier series, etc.

Frequently Asked Questions

What is the Series formula?

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

How do you use the Series formula?

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.

What do the symbols mean in the Series formula?

\sum a_n

Why is the Series formula important in Math?

Foundation for advanced math: Taylor series, Fourier series, etc.

What do students get wrong about Series?

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

What should I learn before the Series formula?

Before studying the Series formula, you should understand: sequence.

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