Infinite Series

Infinite Series

A series that has no last term, such as 1 + 1/4 + 1/9 + 1/16 + ... + 1/n²+ ... . The sum of an infinite series is defined as the limit of the sequence of partial sums.

Note: The infinite series above happens to have a sum of π²/6.

Example showing 0.9 + 0.09 + 0.009 + 0.0009 + … + 9·10⁻ⁿ + … = 1, with partial sums .9, .99, .999, .9999 approaching limit 1.

Key Formula

\sum_{n=1}^{\infty} a_n = \lim_{N \to \infty} S_N = \lim_{N \to \infty} \sum_{n=1}^{N} a_n

Where:

$a_n$ = The nth term of the sequence being summed
$S_N$ = The Nth partial sum, equal to the sum of the first N terms
$N$ = The number of terms included in the partial sum
$\sum_{n=1}^{\infty}$ = Summation notation indicating the sum runs over infinitely many terms

Worked Example

Problem: Find the sum of the infinite geometric series: 1 + 1/2 + 1/4 + 1/8 + ...

Step 1: Identify the first term and common ratio. The first term is a = 1, and each term is multiplied by r = 1/2 to get the next term.

a = 1, \quad r = \frac{1}{2}

Step 2: Write out the first few partial sums to see the pattern.

S_1 = 1, \quad S_2 = 1.5, \quad S_3 = 1.75, \quad S_4 = 1.875

Step 3: Since |r| < 1, this geometric series converges. Apply the infinite geometric series formula.

\sum_{n=0}^{\infty} ar^n = \frac{a}{1 - r}

Step 4: Substitute a = 1 and r = 1/2 into the formula.

\frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2

Answer: The infinite series converges to a sum of 2.

Another Example

This example differs from the first by showing a divergent series. It demonstrates that having terms shrink toward zero is not sufficient for convergence — the terms must shrink fast enough.

Problem: Determine whether the infinite series 1 + 1/2 + 1/3 + 1/4 + ... (the harmonic series) converges or diverges.

Step 1: Write the series in summation notation.

\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots

Step 2: Group terms to establish a lower bound. Notice that 1/3 + 1/4 > 1/4 + 1/4 = 1/2, and 1/5 + 1/6 + 1/7 + 1/8 > 4 × (1/8) = 1/2, and so on. Each grouped block sums to more than 1/2.

1 + \frac{1}{2} + \underbrace{\left(\frac{1}{3} + \frac{1}{4}\right)}_{> \,1/2} + \underbrace{\left(\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}\right)}_{> \,1/2} + \cdots

Step 3: Since you keep adding blocks each worth more than 1/2, the partial sums grow without bound.

S_N \to \infty \text{ as } N \to \infty

Answer: The harmonic series diverges — it has no finite sum, even though its individual terms approach 0.

Frequently Asked Questions

What is the difference between a convergent and divergent infinite series?

A convergent infinite series has partial sums that approach a specific finite number as you include more and more terms. A divergent infinite series does not settle on a finite value — the partial sums may grow without bound, oscillate, or otherwise fail to approach a limit. To find a meaningful "sum," the series must converge.

Can an infinite series have a finite sum?

Yes. When the limit of the partial sums exists and equals a finite number, the infinite series is said to converge to that sum. For example, the geometric series 1 + 1/2 + 1/4 + 1/8 + ... converges to 2. However, not every infinite series has a finite sum — the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges to infinity.

If the terms of a series go to zero, does the series always converge?

No. Having terms approach zero is necessary for convergence but not sufficient. The harmonic series is the classic counterexample: each term 1/n approaches 0 as n grows, yet the series diverges. You must apply a convergence test (such as the ratio test, comparison test, or integral test) to determine whether a series actually converges.

Infinite Series vs. Finite Series

	Infinite Series	Finite Series
Number of terms	Infinitely many terms; no last term	A fixed, finite number of terms
Sum	Defined as the limit of partial sums; may converge or diverge	Always a well-defined finite number (just add the terms)
Notation	$\sum_{n=1}^{\infty} a_n$	$\sum_{n=1}^{N} a_n$
Key question	Does the series converge, and if so, to what value?	What is the value of the sum?
Example	1 + 1/2 + 1/4 + 1/8 + ... = 2	1 + 1/2 + 1/4 = 1.75

Why It Matters

Infinite series appear throughout calculus and beyond — Taylor series let you approximate functions like sin(x), eˣ, and ln(x) as infinite polynomials. In physics, Fourier series decompose periodic signals into infinite sums of sines and cosines, which is fundamental to signal processing and acoustics. Understanding convergence is also essential in probability, where expected values are often computed as infinite sums.

Common Mistakes

Mistake: Assuming a series converges just because its terms approach zero.

Correction: Terms going to zero is necessary but not sufficient. The harmonic series (1 + 1/2 + 1/3 + ...) has terms going to zero yet diverges. Always apply a convergence test.

Mistake: Confusing a sequence with a series.

Correction: A sequence is an ordered list of numbers (e.g., 1, 1/2, 1/4, 1/8, ...). A series is what you get when you add those numbers together (1 + 1/2 + 1/4 + 1/8 + ...). The sequence of terms can converge to zero while the series itself diverges, or the series can converge to a sum entirely different from any individual term.

Related Terms

Series — General concept of summing terms
Sequence of Partial Sums — Defines convergence of an infinite series
Infinite Geometric Series — Most common type of infinite series
Limit — Used to define the sum of the series
Term — Each individual element being summed
Sum — The result of adding terms together
Infinite — Describes the unbounded number of terms
Finite — Contrasts with infinite; finite series end