Series (mathematics)

A series is the sum of all the mathematical objects, or terms, in a sequence, which may be numbers or functions. Series may be finite or infinite. A key aspect of mathematics, they are also important to related fields such as computer science and physics.

Mathematically, computers can really only add, multiply, and compare two numbers. Thus, a natural question is how can a computer evaluate a function such as which cannot be expressed as a finite number of additions and multiplications. Likewise, how can a computer deal with numbers that have a non-terminating decimal expansion?

The answer to such questions lies in infinite series. A series is an infinite sum of the numbers , where ranges from one to infinity. This sum is represented by .

If the numbers get small enough, then successive terms contribute less to the sum. Hence, there is the possibility that the series will converge to a finite number. In this case, we can approximate the value of the sum by looking at the partial sum, for sufficiently large values of . Partial sums are an important tool in determining the convergence of series.

Overview

Two of the more important series are the geometric series and the p-series. The geometric series is of the form . This converges to for and diverges otherwise. The p-series is of the form . This converges for and diverges otherwise. In particular, if , then this is called the harmonic series.

There are a number of tests that can determine the convergence of series that are neither a geometric series nor a p-series. The test for divergence states that if , then the series diverges. The integral test compares the series against the corresponding improper integral. The direct comparison test states that:

If is a convergent series and , then is likewise convergent.

If is a divergent series and , then is likewise divergent.

The limit comparison test states that if , where , then and both converge or both diverge. The ratio test computes then uses this number to determine convergence in the following way:

If , then converges.

If , then diverges.

If , then the test is inconclusive.

-+The root test instead computes

but has the same conclusions as the ratio test. Finally, the alternating series test states that if , , and , then the series . converges.

If converges, then is said to be absolutely convergent. With the exception of the Alternating Series Test, all of the above tests look for absolute convergence. If diverges but converges, then the series is said to be conditionally convergent. Since absolute convergence is preferable to conditional convergence, the Alternating Series Test is usually the last one implemented.

Power Series

Series can be used to approximate functions such as sine, we can employ series. A power series for a function centered at is the function

A power series will converge for all in an interval (which may consist of a single point) and diverge for all outside of . This interval is called the interval of convergence. The radius of convergence is half of the width of this interval. The radius of convergence can usually be computed using the ratio test or the root test. However, neither test will give information about convergence at the endpoints of the interval. Hence, another test must be used for the endpoints.

The coefficients for the above power series can be computed by

where is the kth derivative of evaluated at . This is called the Taylor series for If , then this is called the Maclaurin series. When this method is used on the function this gives the geometric series . Other important power series (and their respective intervals of convergence) include:

Bibliography

Hass, Joel, Maurice Weir, and George B. Thomas. Thomas' Calculus. 13th ed. Boston: Pearson, 2014. Print.

Hirschman, Isadore Isaac. Infinite Series. Belmont: Cengage, 2014. Print.

Knopp, Konrad. Theory and Application of Infinite Series. Mineola: Dover, 1990. Print.

Wade, William R. An Introduction to Analysis. 4th ed.. Boston: Pearson, 2009. Print.