9.1.2Trouble in Paradox

More Infinite Geometric Series

9-18.

Let $S$ be the sum of an infinite geometric series with first term $a$ and common ratio $r$ . In symbols, $S = a + ar + ar^2 + ar^3 + …$ . Work with your team to find a way to demonstrate that $S = \frac { a } { 1 - r }$ . Be prepared to share your method with the class.

9-19.

SOME INFINITE SUM PARADOXES

In the 18^th century, much work was done with infinite series. Instead of infinite sums of numbers, mathematicians worked with “infinite polynomials” such as:

$x - \frac { x ^ { 3 } } { 3 ! } + \frac { x ^ { 5 } } { 5 ! } - \frac { x ^ { 7 } } { 7 ! } + \ldots$

Since they regarded such infinite polynomials as being almost the same as infinite series, they were not prepared to face the unexpected problems that infinite series present. When difficulties arose, however, it caused mathematicians to stop and reconsider.

To illustrate these “problems,” consider the infinite geometric series:

$S = 2 + 4 + 8 + 16 + ...$

Explain why this series is geometric.
Use your method from problem 9-18 to calculate the sum, $S$ .
The sum you computed in part (b) is problematic because the formula $S = \frac { a } { 1 - r }$ will only work if the sum is finite. With your team, discuss ways to restrict the values of $r$ in order to guarantee a finite sum.

9-20.

TROUBLE IN PARADOX

Asa and his teammates are examining the infinite geometric series $S = 1 – 1 + 1 – 1 + 1 – 1 + …$ . They notice that the common ratio is $–1$ . They are wondering if the sum is finite.

Asa grouped the terms like this: $S = (1 – 1) + (1 – 1) + (1 – 1) + …$ What value does he get for $S$ ?
Bianca grouped her terms this way: $S = 1 – (1 – 1) – (1 – 1) – (1 – 1) + …$ What value does she get for $S$ ?
Charles has a different idea. He uses the formula $S = \frac { a } { 1 - r }$ , hoping not to get a negative value. What value does he get for $S$ ?
Based on your observation, does the sum exist? Explain.

Math Notes

Sum of Infinite Geometric Series

Every series has two associated sequences, the sequence of terms and the sequence of partial sums. For example, the series $\frac { 1 } { 2 } , \frac { 1 } { 4 } , \frac { 1 } { 8 } , \frac { 1 } { 16 } ,$ is made from the sequence of terms $\frac { 1 } { 2 } , \frac { 1 } { 4 } , \frac { 1 } { 8 } , \frac { 1 } { 16 } ,$ , which can be written $u _ { n } = \frac { 1 } { 2 ^ { n } }$ . We use this $n$ ^th term when we write the associated series using sigma notation: $\displaystyle \sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 ^ { n } }$ .

We can use this series to generate a different sequence: the sequence of partial sums. Its first four terms are:

$S _ { 1 } = \frac { 1 } { 2 }$ , $S _ { 2 } = \frac { 1 } { 2 } + \frac { 1 } { 4 } = \frac { 3 } { 4 }$ _, $S _ { 3 } = \frac { 1 } { 2 } + \frac { 1 } { 4 } + \frac { 1 } { 8 } = \frac { 7 } { 8 }$ , and $S _ { 4 } = \frac { 1 } { 2 } + \frac { 1 } { 4 } + \frac { 1 } { 8 } + \frac { 1 } { 16 } = \frac { 15 } { 16 }$

Note that $u_{n}$ and $S_{n}$ are different: $u_{n}$ is just the $n$ ^th term of the sequence of terms, while $S_{n}$ is the sum of the first $n$ terms of $u_{n}$ . The sum of an infinite series is the limit of its sequence of partial sums if the limit is finite. In symbols, $\lim\limits _ { n \rightarrow \infty } S _ { n }$ .

In this example, $u _ { n } = \frac { 1 } { 2 ^ { n } }$ and S = $\frac { 1 } { 2 } , \frac { 1 } { 4 } , \frac { 1 } { 8 } , \frac { 1 } { 16 } ,$ $=\lim\limits _ { n \rightarrow \infty } S _ { n }=1$ , where the sequence of partial sums is $S _ { n } = \frac { 1 } { 2 } , \frac { 3 } { 4 } , \frac { 7 } { 8 } , \frac { 15 } { 16 } , \ldots , \frac { 2 ^ { n } - 1 } { 2 ^ { n } }$ . This agrees with the result of the formula $S = \frac { a } { 1 - r }$ .

9-21.

Consider the infinite geometric series $1 + 2 + 4 + 8 +…$ . Write a sequence of the first four partial sums of this series, as shown in the preceding Math Notes box. Then, use the limit of the sequence as $n → ∞$ to prove that this series does not have a finite sum.

9-22.

Though cumbersome, determining the limit of the sequence of partial sums is useful because, unlike using the formula $S = \frac { a } { 1 - r }$ , it reveals series that fail to have one unique sum. For example, recall the series $1 – 1 + 1 – 1 + 1 – 1 +…$ from problem 9-20. The first four partial sums of this series are:

$S_1 = 1$

$S_2 = 1 – 1 = 0$

$S_3 = 1 – 1 + 1 = 1$

$S_4 = 1 – 1 + 1 – 1 = 0$

Write a rule for the sequence of partial sums, $S_n$ , in terms of $n$ .
Evaluate $\lim\limits _ { n \rightarrow \infty } S _ { n }$ . Explain why this limit demonstrates that the sum of the series does not exist.

9-23.

If $xy + \ln(y) = 2$ , determine the exact value of $\frac { d y } { d x }$ when $y = 1$ . Homework Help ✎

9-24.

Betsy accelerated quickly from a stop sign at the freeway entrance. A table of her velocity is shown below. Approximate how far she went in the first $10$ seconds using a trapezoidal approximation with subintervals of $dt = 2$ seconds. Homework Help ✎

Time (sec)	Vel. (ft/sec)
$0$	$0$
$2$	$19$
$4$	$36$
$6$	$52$
$8$	$70$
$10$	$84$

9-25.

A square plot of land $100$ feet on each side is shown in the diagram at right. A pipe is to be laid in a straight line from point $A$ to a point $P$ on side $\overline{BC}$ , and from there to point $C$ . (Point $P$ cannot be point $B$ or point $C$ .) The cost of laying the pipe is $$20$ per foot if the pipe goes through the lot from point $A$ to point $P$ (since it must be laid underground) and $$10$ per foot if it is laid along one of the sides of the lot. What is the most economical way to lay the pipe? Homework Help ✎

Square, A B C D, with point, P, on side, B C, about 1 fourth of the way from B to C, bold segments from, A to P, & from P to C.

9-26.

If a hole is punched in the bottom of a tall open can, the rate at which liquid leaks out is roughly proportional to the square root of the depth of the remaining liquid. Unfortunately, Jose’s pesky little brother just used a pen to punch a hole in the bottom of his $16$ cm tall (completely full) can of soda, decreasing the depth of the precious liquid at the rate of $12$ cm/min. What is the height of the soda in the can after $20$ seconds? Homework Help ✎

9-27.

No calculator! Write and evaluate an integral expression to calculate the volume generated when the region bounded by $f(x) = |x|$ and $g(x) = x^2$ is rotated about the $y$ -axis. Homework Help ✎

Compute without a calculator

9-28.

At right is a graph of $y = f(t)$ . If $F(x)=\int_1^xf(t)dt$ then determine: Homework Help ✎

$F(0)$

$F(1)$

$F^\prime(3)$

$F(6)$

Continuous Linear piecewise labeled, f of t, starting at the point (0, comma 2), turning down @ (1, comma 5), up @ (4, comma negative 1), down @ (6, comma 1), ending @ (7, comma 0).

9-29.

No calculator! Find the interval(s) over which $f(x)=\frac{2}{x^2+1}$ is concave down. Homework Help ✎

Compute without a calculator

9-30.

CALCULATING PARTIAL SUMS FOR AN INFINITE GEOMETRIC SERIES

We can use the method for determining the sum of an infinite series to calculate finite partial sums. Finish the work below. Your end result should be $S _ { n } = \frac { a - a r ^ { n } } { 1 - r }$ . Homework Help ✎

$\left. \begin{array} \\ { S _ { n } = a + a r + a r ^ { 2 } + \ldots + a r ^ { n - 1 } } \\ { r S _ { n } = \quad a r + a r ^ { 2 } + \ldots + a r ^ { n - 1 } + a r ^ { n } } \end{array} \right.$

9-31.

An alternating series, such as $100 – 80 + 64 – 51.2 + …$ , has terms that alternate in sign ( $+$ and $–$ ). Homework Help ✎

What is the common ratio for $100 – 80 + 64 – 51.2 + …$ ?
Calculate the sum of this infinite series.

Calculus Third Edition

Sum of Infinite Geometric Series