Series Tests

In mathematics, especially in real analysis, the study of sequences and series is a fundamental cornerstone. A series can be thought of as the sum of a sequence of numbers. However, unlike finite sums, series contain possibly infinite numbers. Evaluating a series involves determining whether the sum converges to a finite value. This study of series convergence is done through various series tests.

Comprehension series

A series is defined as the sum of the terms of a sequence. If we have a sequence of numbers a_1, a_2, a_3, ldots then the corresponding series is

S_n = a_1 + a_2 + a_3 + ldots + a_n

As n becomes very large. The series is said to be convergent if the sequence of partial sums S_n approaches a finite limit as n goes to infinity.

For example, consider the series formed by the sequence 1, 1/2, 1/4, 1/8, ldots. The nth partial sum is:

S_n = 1 + frac{1}{2} + frac{1}{4} + ldots + frac{1}{2^{n-1}}

As a visual example:

This series can be recognized as a geometric series, and it converges to 2.

Series testing

To determine the convergence or divergence of a series, a number of tests are typically used. Common and widely used tests in real analysis include:

n-term test

The nth-term test involves analyzing the limit of the nth-term of the series. If lim (a_n) != 0, then the series sum a_n diverges. Although this test is quite easy to apply, it is not conclusive for convergent series.

Example:

a_n = n

Here, lim (a_n) = ∞, and thus, the series sum n diverges.

Geometric series test

The geometric series is a series of the form

a + ar + ar^2 + ar^3 + ldots

The series converges only when the absolute value of the common ratio r is less than 1. Specifically, the sum of the series is a / (1 - r) when |r| < 1.

Example:

1 + frac{1}{2} + frac{1}{4} + frac{1}{8} + ldots

This series has a = 1 and r = 1/2. Since |r| < 1, the series converges, and its sum is 1 / (1 - 1/2) = 2.

P-series testing

The format of the P-series is as follows

sum frac{1}{n^p}

If p > 1 then it converges and if p <= 1 then it diverges.

Example:

1. p = 2 :

   sum frac{1}{n^2}

   It converges.

2. p = 1 :

   sum frac{1}{n}

   divergence (known as the harmonic series).

Ratio test

The ratio test examines the behaviour of ratios of consecutive terms. Given a series sum a_n, calculate

L = lim_{n to infty} left| frac{a_{n+1}}{a_n} right|

The test says:

• If L < 1, then the series converges absolutely.
• If L > 1 or L is infinite, the series diverges.
• If L = 1, the test is inconclusive.

Example:

a_n = frac{1}{n!}

Here, L = lim_{n to infty} left(frac{1}{(n+1)!} / frac{1}{n!}right) = lim_{n to infty} frac{1}{n+1} = 0. Since L < 1, the series sum frac{1}{n!} converges absolutely.

Original test

Also known as the Cauchy root test, it tests the limit:

L = lim_{n to infty} sqrt[n]{|a_n|}

The basic test follows the same rules:

• If L < 1, then the series converges absolutely.
• If L > 1 or L is infinite, the series diverges.
• If L = 1, the test is inconclusive.

Integral test

For a series sum a_n where a_n = f(n) for some positive, decreasing and continuous function f on [1, infty), the integral test can be applied.

This test determines convergence by comparing the series to an improper integral

int_1^infty f(x), dx

• If int_1^infty f(x), dx converges, then the series converges.
• If int_1^infty f(x), dx diverges, then so does the series.

Example:

sum frac{1}{n^2}

Using integrals

int_1^infty frac{1}{x^2}, dx

Solving int_1^infty frac{1}{x^2}, dx = 1, which converges, and the series also converges.

Alternating series test

For alternating series of the form

sum (-1)^n b_n

Where b_n is positive, the series converges if:

• The sequence b_n is monotonically decreasing, that is, b_{n+1} <= b_n.
• lim_{n to infty} b_n = 0

Limit comparison test

Two series sum a_n and sum b_n are given with positive terms, if

L = lim_{n to infty} frac{a_n}{b_n}

and 0 < L < infty, then both series will converge or both will diverge.

For example, comparing sum frac{1}{n^2} with sum frac{1}{n^3}, it is known that sum frac{1}{n^3}, the p-series with p = 3 > 1, converges. Calculating the limit:

L = lim_{n to infty} frac{1/n^2}{1/n^3} = lim_{n to infty} n = infty

Since L emerges as infty after simplification, this test is not useful in finding convergence.

Conclusion

Series tests are indispensable tools for analyzing infinite series. These tests provide criteria for determining whether a given series exhibits convergence, divergence, or whether a decision requires additional analysis. By applying these tests, one can systematically look at series in various mathematical domains, paving the way for deeper analytical insights. Understanding their use comes not only from knowing their definitions, but also from recognizing their applicability through examples and exercises.

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