Find the sum of the infinite seriesif possible. We will use the formula for the sum of infinite geometric sequence,to help us with this problem.

This is primarily a list of Greatest Mathematicians of the Past, but I use birth as an arbitrary cutoff, and two of the "Top " are still alive now. Click here for a longer List of including many more 20th-century mathematicians.

Click for a discussion of certain omissions. Please send me e-mail if you believe there's a major flaw in my rankings or an error in any of the biographies. Obviously the relative ranks of, say Fibonacci and Ramanujan, will never satisfy everyone since the reasons for their "greatness" are different.

I'm sure I've overlooked great mathematicians who obviously belong on this list. Please e-mail and tell me! Following are the top mathematicians in chronological birth-year order. By the way, the ranking assigned to a mathematician will appear if you place the cursor atop the name at the top of his mini-bio.

Earliest mathematicians Little is known of the earliest mathematics, but the famous Ishango Bone from Early Stone-Age Africa has tally marks suggesting arithmetic.

The markings include six prime numbers 5, 7, 11, 13, 17, 19 in order, though this is probably coincidence. The advanced artifacts of Egypt's Old Kingdom and the Indus-Harrapa civilization imply strong mathematical skill, but the first written evidence of advanced arithmetic dates from Sumeria, where year old clay tablets show multiplication and division problems; the first abacus may be about this old.

By years ago, Mesopotamian tablets show tables of squares, cubes, reciprocals, and even logarithms and trig functions, using a primitive place-value system in base 60, not Babylonians were familiar with the Pythagorean Theorem, solutions to quadratic equations, even cubic equations though they didn't have a general solution for theseand eventually even developed methods to estimate terms for compound interest.

The Greeks borrowed from Babylonian mathematics, which was the most advanced of any before the Greeks; but there is no ancient Babylonian mathematician whose name is known.

Also at least years ago, the Egyptian scribe Ahmes produced a famous manuscript now called the Rhind Papyrusitself a copy of a late Middle Kingdom text. It showed simple algebra methods and included a table giving optimal expressions using Egyptian fractions.

Today, Egyptian fractions lead to challenging number theory problems with no practical applications, but they may have had practical value for the Egyptians.

The Pyramids demonstrate that Egyptians were adept at geometry, though little written evidence survives. Babylon was much more advanced than Egypt at arithmetic and algebra; this was probably due, at least in part, to their place-value system.

But although their base system survives e. The Vedics understood relationships between geometry and arithmetic, developed astronomy, astrology, calendars, and used mathematical forms in some religious rituals.

The earliest mathematician to whom definite teachings can be ascribed was Lagadha, who apparently lived about BC and used geometry and elementary trigonometry for his astronomy. Apastambha did work summarized below; other early Vedic mathematicians solved quadratic and simultaneous equations.

Other early cultures also developed some mathematics. The ancient Mayans apparently had a place-value system with zero before the Hindus did; Aztec architecture implies practical geometry skills.

Ancient China certainly developed mathematics, in fact the first known proof of the Pythagorean Theorem is found in a Chinese book Zhoubi Suanjing which might have been written about BC. Thales may have invented the notion of compass-and-straightedge construction. Several fundamental theorems about triangles are attributed to Thales, including the law of similar triangles which Thales used famously to calculate the height of the Great Pyramid and "Thales' Theorem" itself: The other "theorems" were probably more like well-known axioms, but Thales proved Thales' Theorem using two of his other theorems; it is said that Thales then sacrificed an ox to celebrate what might have been the first mathematical proof in Greece.

Thales was also an astronomer; he invented the day calendar, introduced the use of Ursa Minor for finding North, invented the gnomonic map projection the first of many methods known today to map part of the surface of a sphere to a plane, and is the first person believed to have correctly predicted a solar eclipse.We will do more like this in the Writing Formulas section below..

Summary of Formulas for Sequences and Series. Before we get started, here is a summary of the main formulas for Sequences and Series. Geometric Series and Convergence Theorems.

Geometric Series and Convergence Theorems. We begin this section by presenting a series of the form, which then the series converges absolutely if and diverges if. Proof.

Example Show that converges. Solution. Using the ratio test, we find that.

Geometric Series. A geometric series is any series that can be written in the form, \[\sum\limits_{n = 1}^\infty {a{r^{n - 1}}} \] or, with an index shift the geometric series will often be written as, \[\sum\limits_{n = 0}^\infty {a{r^n}} \] These are identical series and will .

Practice writing finite geometric series like 3 + 6 + 12 + 24 in sigma notation.

Start studying 1 Sequences & Series. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Practice writing finite geometric series like 3 + 6 + 12 + 24 in sigma notation.

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Read Solutions to Time Series Analysis With Applications in R, second edition