Series and Convergence

image of Sonya Kovalevsky Sonya Kovalevsky (1850-1891), also known as Sofia Kovalevskaya, was the daughter of Russian nobility who displayed a remarkable aptitude for mathematics at an early age and learned trigonometry and calculus on her own. Because her father would not allow her to leave home to study at a university, she left Russia at age 18 to study in Germany where she was not allowed to attend lectures officially. She worked with the mathematician Karl Weierstrass, received her doctorate at age 24 from Göttingen University, and wrote remarkable research papers on infinite series, partial differential equations, and Abelian integrals.

Infinite Series and Partial Sums

Adding the terms of an infinite sequence equation image indicator $LaTeX: \left\{a_n\right\}_{n=1}^{\infty}$ produces an expression of the form a₁ + a₂ + a₃ + ... + a_n + ..., which is called an infinite series or more simply a series. The notation used to represent an infinite series is $LaTeX: \sum_{n=1}^{\infty}a_n\:or\:\sum a_n$ . Because this series contains infinitely many terms, calculating the sum through continual addition of subsequent terms is impossible. However, it is possible to add terms one at a time from the beginning, look at each resulting sum, and search for some pattern in each partial sum. Consider the partial sums s_i defined by

s₁ = a₁

s₂ = a₁ + a₂

s₃ = a₁ + a₂ + a₃_....

s_n = a₁ + a₂ + a₃ +...+ a_n = equation image indicator $LaTeX: \sum_{i=1}^na_'$

The number s_n is called the nth partial sum, and these partial sums form a sequence {s_n}, which may or may not have a limit.

The two examples below use a graph of a finite number of partial sums to assist in predicting whether or not a limiting sum value exists.

graph of partial sums of infinite series getting smaller

In the first graph the partial sums appear to approach a limiting sum of 2, whereas in the second graph, the partial sums are constantly increasing and do not appear to approach a limiting sum.

Depending on the number and nature of the terms to be summed, it may be helpful to use technology to assist in the computation. The next presentation outlines how to use the TI-84Plus to determine the sum of a finite number of terms in a series.

Convergent and Divergent Series

If the sequence of partial sums s_n = a₁ + a₂ + a₃ + ... + a_n = equation image indicator $LaTeX: \sum_{i=1}^na_'$ converges to a real number limit L, the series converges or is convergent and its sum is L. If the sequence of partial sums does not converge, then the series diverges or is divergent. The presentation below explores how partial sums are used to investigate whether or not a series converges or diverges.

Additional investigations to determine the convergence or divergence of several infinite series are found below. View the beginning until 5:31.

Geometric Series

You may recall from precalculus that a series of the form S = a + ar + ar² + ... + ar^n-1, where r is the nonzero constant ratio between any two consecutive terms and a ≠ 0, is a finite geometric series. You may also recall the formula for the sum of a geometric series containing a finite number of terms, equation image indicator $LaTeX: S_n=\frac{a_{1\left(1-r^n\right)}}{1-r}$ . Derivation of this formula for the sum of a finite geometric series is the focus of the next presentation.

In this course a geometric series refers to an infinite geometric series whose definition is very similar to the finite geometric series definition. An infinite geometric series is defined by S = a + ar + ar² + ... + arⁿ^-1 + ... = equation image indicator $LaTeX: \sum_{n=1}^{\infty}ar^{n-1}$ , where r is the nonzero constant ratio between any two consecutive terms and a ≠ 0. The sum of an infinite geometric series is given by $LaTeX: \sum_{n=1}^{\infty}ar^{n-1}=\frac{a}{1-r}$ , where |r| < 1. The geometric behavior of an infinite geometric series is the focus of the next presentation.

Applications of Geometric Series

Two classic applications of infinite geometric series are repeating decimals and the bouncing ball problem. Examples of these applications are features in the presentation below.

Harmonic Series

An infinite series in which the reciprocals of the terms form the arithmetic sequence 1 + 1/2 + 1/3 + 1/4 + ... = equation image indicator $LaTeX: \sum_{n=1}^{\infty}\frac{1}{n}$ is the harmonic series. Using the TI-84Plus to graph partial sums of the harmonic series helps to develop an intuitive understanding of whether or not the harmonic series converges or diverges. View the presentation below from the beginning to 3:40.

image of Nicole Oresme Image Caption: Nicole Oresme (1320-1325), a 14^th century philosopher, developed the first proof of the divergence of the harmonic series - something that was only replicated in the later centuries. His proof, an alternative to other "standard" tests for divergence, elegantly stated that for any value of 1/n, the closest n that is a member of the sequence 2ⁿ, the preceding n/2 terms must be greater than 1/2. Therefore, by using the Comparison Test and the Squeeze Theorem, the series must be greater than the series 1 + 1/2 + 1/2 + 1/2 + ... + 1/2, which means the harmonic series whose terms are 1/n must be divergent. Oresme was the only mathematician to prove this fact, and held that honor for the next few centuries.

Using the method of partial sums, it can be shown analytically that the harmonic series is divergent. See below.

examples of divergent Harmonic Series

Telescopic Series

Any series where nearly every term cancels with a preceding or following term is a telescoping series. Consider the series equation image indicator $LaTeX: \sum_{n=1}^{\infty}=\left(\frac{1}{n}-\frac{1}{n+1}\right)=\left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+...$ . The nth partial sum of the series is $LaTeX: s_n=1-\frac{1}{n+1}$ and because the limit of s_n is 1, the series converges and its sum is 1. The next presentation features other examples of telescoping series.

nth Term Test for Divergence

The nth Term Test for Divergence states that if equation image indicator $LaTeX: \lim_{n \to \infty} a_n$ does not exist or if $LaTeX: \lim_{n \to \infty} a_n \ne 0$ , then $LaTeX: \sum_{n=1}^{\infty }a_n$ diverges. Note that this test is never used to determine convergence.

To explore the convergence or divergence for various series using an interactive applet, CLICK HERE.

Expressing the nth-Term Test for Divergence using the equivalent contrapositive form, produces the following useful theorem:

If the series equation image indicator $LaTeX: \sum_{n=1}^{\infty }a_n$ is convergent, then $LaTeX: \lim_{n \to \infty} a_n = 0$ .

Series and Convergence Practice

Series and Convergence: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of series, convergence, and divergence.

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