reset x

Perform the following multiplications

Use expand to compute:

1)

2)

3)

What do you suppose is the following product?

Use s implify to compute:

4)

5)

In this case we need to define x and n. Try experimenting with values of x and n to determine partial sums.

Must n be a positive integer?

a) Try various values of n. How many different n's do you need to see a pattern?

b) What happens if as n gets large? Make a conjecture, (a guess).

c) What happens if as n gets large? Make a conjecture.

Find an infinite series for the shaded area in the diagram below.

Why must the series converge?

What is it's sum?

Given a geometric series, we can answer the Basic Questions for Infinite Series. As you will discover, it is often very difficult to find the sum of an infinite series.

Convergence of Geometric Series

Our experiments above gave us some insights about the convergence of geometric series. We can use some mathematical analysis to make our conjectures more precise. The algebra will reveal the underlying pattern.

so we have found a closed form for the sum of any geometric series when :

The Sum of a Geometric Series

Two Basic Questions for Infinite Series

1. Does the Series converge?

2. What is it's Sum?

Assignment:

Find the sum of the series:

a)

b)

c) Gabriel's Staircase:

if r = 1/2 does the series converge?

d) The Rice and Chessboard Problem. Let one grain of rice be placed on the first square of an 8 x 8 chessboard , two on the second, four on the third, eight on the fourth, etc. How many grains total are placed on an 8 x 8 chessboard? (Find S64)

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Use these patterns to find:

What does this equal?

We notice that for finite values of n,

What happens as n goes to infinity?

For values of what is ?

Experiment with these values.

As n gets large, the partial sums converge to the limit:

for

Given

Eq. 1)

Replace the 1 with , replace the with , the , , with , etc. Next observe the lessor series can be written as:

This series must diverge since it is

the repeated addition of .

The above examples lead us to ask the question, how do we know if an infinite series converges or diverges, and if it converges, what is the sum?

Oresme's proof of the divergence of the harmonic series was lost, and later reproved by Johan Bernoulli. Filled with his success he tried to find the sum of the following :

Practice: Bernoulli's Problem

Given the series

Eq 2)

1. Does the series converge or diverge?

2. If it converges, what is it's sum?

An Introduction to Infinite Series

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Introduction

An infinite series is simply an infinite sum of numbers. The harmonic series , formed by summing the reciprocals of the natural numbers is a good example of an infinite series:

The decimal fraction for could be thought of as the sum of the infinite series

Infinite Series in Antiquity

Archimedes and the Quadrature of a Parabolic Segment

As far back as the Greeks, mathematicians have wrestled with infinite series. Archimedes of Syracuse (c. 287 - 212 B.C.) in his treatise Quadrature of the Parabola , calculated the area of a parabolic sector as a sum of the areas of inscribed triangles. He began by constructing the inscribed triangle D PQq and calculating its area, then continued by constructing the triangles DPRQ and DPrq and calculating their areas. Continuing in this manner he discovered the pattern,

where is the area of D PQq and is the area of the triangles DPRQ and DPrq, and the higher powers of are the areas of the triangles

Quadrature of the Parabola

formed by filling in the remaining are with more and more triangles. Archimedes correctly found the sum of this series was .. That an infinite sum could add up to a finite number was a paradox for the Greeks.

Oresme and the Sum of the Harmonic Series

Oresme (c. 1360) in his tract, Quaestiones Super Geometrium Euclidus , proved that the harmonic series is divergent, that is the series does not add up to any finite number! His proof was to compare the harmonic series with a series of lesser terms:

Partial Sums of an Infinite Series

As you attempted the Bernoulli problem above, you were likely to begin by adding up the first few terms of the series just to get a feel for what happens. Adding up the first n terms of a series is what is referred to as the nth partial sum, and is denoted as Sn.

If we continue adding terms, will the partial sums converge to a limit?

Compute the partial sums.

As you compute the partial sums, look for patterns!

The nth partial sum of the geometric series above is:

Try changing the value of n. Do you observe any patterns?

What is the fraction form for the 103rd partial sum?

What is

?

Exploration:

Compute the partial sums for various values of x

Given:

Geometric Series

Consider the infinite series given by:

Eq. 3)

This infinite series is an example of what is called a geometric series or geometric progression. Does this series converge? What is it's limit?

Sometimes a picture helps.

Geometric series are a good starting point for any introduction to infinite series. They are easy to work with and provide good illustrations of the basic concepts of infinite series.

Compare the following two series.

If the second series converges, why must the first series also converge?

The Definition of a Geometric Series

A series of the form is called a geometric series. Notice that the ratio of two consecutive terms of the series, .

If a = 1 the geometric series looks like: