Infinite Sums and Products

Infinity, as an informal concept, is associated with endless repetition. Common expressions like "and so on" and "etcetera" and, occasionally, the ellipses "..." reflect a concept that mathematics attempts to make rigorous. On this page I shall collect a few appealing formulas whose meaning I hope will be intuitively clear even without formal justification. I believe this could be a good way to illustrate one of the uses of infinity in mathematics.

Infinite sums

1. Geometric series: ∑_n≥02^-n

   20 + 2-1 + 2-2 + 2-3 + ... = 2.

   
   

2. Telescoping series: ∑_n≥11 / n(n+1)

   1 / 1·2 + 1 / 2·3 + 1 / 3·4 + 1 / 4·5 + ... = 1.

   
   

3. James Gregory's series

   1/1 - 1/3 + 1/5 - 1/7 + ... = π/4.

   
   

4. Euler's series: ∑_n≥1n^-2

   1/1² + 1/2² + 1/3² + 1/4² + ... = π²/6.

Infinite products

1. John Wallis' product

   2·2·4·4·6·6· ... / 1·3·3·5·5·7· ... = π/2.

   
   

2. François Viète's product

   √2/2 · √2 + √2/2 · √2 + √2 + √2/2 ... = 2/π.

   
   

3. ∏_n≥2(1 - n^-2)

   (1 - 1/2²)·(1 - 1/3²)·(1 - 1/4²)· ... = 1/2.

   
   

4. ∏_n≥3(1 - 4n^-2)

   (1 - 4/3²)·(1 - 4/4²)·(1 - 4/5²)· ... = 1/6.

   
   

Continued fractions

1. Golden ratio

   1 + 1/(1 + 1/(1 + 1/(1 + ... ))) = φ = (√5 + 1)/2.

   
   

(For more, check the page on continued fractions.)

What not

1. Golden ratio

   √1 + √1 + √1 + √1 + ... = φ = (√5 + 1)/2.

   
   

(For more, check the page on continued fractions.)

• What Is Infinity?.
• Infinity As a Limit.
  • Infinite Sums and Products
• Cardinal Numbers.
  • Cantor-Bernstein-Schroeder theorem.
• Ordinal Numbers.
  • Order. Well-ordered sets.
• Surreal Numbers.
• Infinitesimals. Non-standard Analysis.
• Various Geometric Infinities
  • Projective Infinity.
  • Infinity in Inversive Geometry.
  • Infinity in Compactification.
• Paradoxes of Infinity
  • Achilles and the Tortoise
  • Infinity and Probability
  • Infinite Border, Finite Area