Infinite Series

Summary: A sum of infinitely many terms that may nonetheless equal a specific finite number; introduced in Elements of Algebra through the repeated long division of fractions.

Sources: chapter-1.2.5, chapter-1.2.12, chapter-1.2.13

Last updated: 2026-04-29

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Definition

An infinite series is an expression of the form

$s_0+s_1+s_2+s_3+\cdots$

where the terms continue without end. Euler establishes that such a series can have a determinate value even though the addition never finishes. (source: chapter-1.2.5)

Origin: Long Division of Fractions

Euler’s route to infinite series is computational: dividing a numerator by a denominator that does not divide it exactly produces a quotient plus a remainder, and dividing the remainder again extends the quotient. Repeating indefinitely, the remainder decreases (when $|a|<1$) and the finite partial sums approach the original fraction. See ch1.2.5-infinite-series. (source: chapter-1.2.5)

Geometric Series

The foundational example is the geometric series in ratio $a$:

$\frac{1}{1-a}=1+a+a^2+a^3+a^4+\cdots$

Every term is $a$ times the previous one. The partial sum of the first $n$ terms is $\frac{1-a^n}{1-a}$, and the residual fraction $\frac{a^n}{1-a}$ vanishes when $|a|<1$. (source: chapter-1.2.5)

The alternating version arises from $1/(1+a)$:

$\frac{1}{1+a}=1-a+a^2-a^3+\cdots$

(source: chapter-1.2.5)

Convergence and Divergence

Euler observes the two regimes without naming them formally:

Condition	Behaviour	Example
$\Vert a\Vert <1$	Partial sums approach the fraction; error shrinks	$a=\frac{1}{2}$: sums approach 2
$a=1$	Series diverges to $\infty$; equals $1/0=\infty$	$1+1+1+\cdots$
$\Vert a\Vert >1$	Series diverges, but remains equal to the fraction if the remainder term is kept	$a=2$: $1+2+4+\cdots +64-128=-1$

(source: chapter-1.2.5)

Grandi’s Series

Setting $a=1$ in $\frac{1}{1+a}=1-a+a^2-\cdots$ gives the famous result:

$\frac{1}{2}=1-1+1-1+1-\cdots$

Euler’s argument: since the series never stops at $+1$ or $-1$, its sum lies between 0 and 1, hence $\frac{1}{2}$. This is an early example of assigning a value to a divergent series by symmetry — later formalised as Cesàro summation. (source: chapter-1.2.5)

General Formula

For any fraction $\frac{c}{a+b}$:

$\frac{c}{a+b}=\frac{c}{a}-\frac{bc}{a^2}+\frac{b^{2}c}{a^3}-\frac{b^{3}c}{a^4}+\cdots$

This is the generalised geometric series with ratio $-b/a$. (source: chapter-1.2.5)

Significance

Euler closes the chapter by noting that the ability to resolve fractions into infinite series has produced “inventions of the utmost importance.” This foreshadows power series expansions of functions, Taylor series, and the use of generating functions — all central to later analysis. (source: chapter-1.2.5)

Binomial Series (Irrational and Negative Powers)

From Chapters 1.2.12 and 1.2.13, the geometric series is a special case ($n=-1$) of the binomial series: substituting any real exponent $n$ into the binomial-theorem formula yields an infinite series for $(a+b{)}^n$:

$$(a+b{)}^n=a^n+\frac{n}{1}{a}^{n-1}b+\frac{n(n-1)}{1\cdot 2}{a}^{n-2}{b}^2+\cdots$$

Key instances:

• $n=\frac{1}{2}$: series for $\sqrt{a+b}$, used to approximate irrational square roots by rationals. (source: chapter-1.2.12)
• $n=\frac{1}{3}$: series for $\sqrt[3]{a+b}$. (source: chapter-1.2.12)
• $n=-1$: recovers the geometric series from Chapter 1.2.5. (source: chapter-1.2.13)
• $n=-2,-3,-4,\dots$: series for $1/(a+b{)}^m$ with natural-number and figurate-number coefficients. (source: chapter-1.2.13)

See ch1.2.12-irrational-powers-infinite-series and ch1.2.13-negative-powers-infinite-series.

Related pages

• ch1.2.5-infinite-series
• ch1.2.12-irrational-powers-infinite-series
• ch1.2.13-negative-powers-infinite-series
• binomial-theorem
• fractions
• infinity
• powers-and-exponents