Answer:
Leonhard Euler was a Swiss mathematician who lived from 1707-1783. He made significant contributions to many areas of mathematics, including number theory, calculus, and geometry.
One of Euler's significant contributions to the field of sequences and series is his work on the summation of infinite series. In particular, he developed a method for finding the sum of an infinite geometric series, which has the form:
a + ar + ar^2 + ar^3 + ...
where a is the first term, r is the common ratio, and the series continues infinitely.
Euler's method for finding the sum of this series is as follows:
If |r| < 1, then the sum of the series is:
a / (1 - r)
For example, let's consider the infinite geometric series:
2 + 4 + 8 + 16 + ...
In this series, a = 2 and r = 2. Since |r| < 1, we can use Euler's formula to find the sum:
sum = a / (1 - r)
= 2 / (1 - 2)
= -2
Therefore, the sum of the infinite geometric series 2 + 4 + 8 + 16 + ... is -2.
Euler's formula is just one example of his many contributions to mathematics. His work on sequences and series laid the foundation for many important concepts in calculus, and his ideas continue to influence the field of mathematics today.