 infinite series

Math.a sequence of numbers in which an infinite number of terms are added successively in a given pattern; the sequence of partial sums of a given sequence.[17901800]
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In mathematics, the sum of infinitely many numbers, whose relationship can typically be expressed as a formula or a function.An infinite series that results in a finite sum is said to converge (see convergence). One that does not, diverges. Mathematical analysis is largely taken up with studying the conditions under which a given function will result in a convergent infinite series. Such series (e.g., the Fourier series) are particularly useful in solving differential equations.* * *
the sum of infinitely many numbers related in a given way and listed in a given order. Infinite series are useful in mathematics and in such disciplines as physics, chemistry, biology, and engineering.For an infinite series a_{1} + a_{2} + a_{3} +⋯, a quantity s_{n} = a_{1} + a_{2} +⋯+ a_{n}, which involves adding only the first n terms, is called a partial sum of the series. If s_{n} approaches a fixed number S as n becomes larger and larger, the series is said to converge (convergence). In this case, S is called the sum of the series. An infinite series that does not converge is said to diverge. In the case of divergence, no value of a sum is assigned. For example, the nth partial sum of the infinite series 1 + 1 + 1 +⋯ is n. As more terms are added, the partial sum fails to approach any finite value (it grows without bound). Thus, the series diverges. An example of a convergent series isAs n becomes larger, the partial sum approaches 2, which is the sum of this infinite series. In fact, the series 1 + r + r^{2} + r^{3} +⋯ (in the example above r equals 1/2) converges to the sum 1/(1 − r) if 0 < r < 1 and diverges if r ≥ 1. This series is called the geometric series with ratio r and was one of the first infinite series to be studied. Its solution goes back to Zeno Of Elea's paradox involving a race between Achilles and a tortoise (see mathematics, foundations of: Being versus becoming (mathematics, foundations of)).Certain standard tests can be applied to determine the convergence or divergence of a given series, but such a determination is not always possible. In general, if the series a_{1} + a_{2} +⋯ converges, then it must be true that a_{n} approaches 0 as n becomes larger. Furthermore, adding or deleting a finite number of terms from a series never affects whether or not the series converges. Furthermore, if all the terms in a series are positive, its partial sums will increase, either approaching a finite quantity (converging) or growing without bound (diverging). This observation leads to what is called the comparison test: if 0 ≤ a_{n} ≤ b_{n} for all n and if b_{1} + b_{2} +⋯ is a convergent infinite series, then a_{1} + a_{2} +⋯ also converges. When the comparison test is applied to a geometric series, it is reformulated slightly and called the ratio test: if a_{n} > 0 and if a_{n + 1}/a_{n} ≤ r for some r < 1 for every n, then a_{1} + a_{2} +⋯ converges. For example, the ratio test proves the convergence of the seriesMany mathematical problems that involve a complicated function can be solved directly and easily when the function can be expressed as an infinite series involving trigonometric functions (sine and cosine). The process of breaking up a rather arbitrary function into an infinite trigonometric series is called Fourier analysis or harmonic analysis and has numerous applications in the study of various wave phenomena.* * *
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