Infinite series (damn mickey mouse)

In summary, an infinite series is a sum of an infinite number of terms that are added together. This is in contrast to a finite series, which has a limited number of terms. The convergence of an infinite series refers to whether or not the sum of the terms is a finite number. Tests such as the ratio test and the integral test can be used to determine the convergence or divergence of an infinite series. Infinite series have many real-world applications in fields such as physics, engineering, finance, and signal processing, where they are used to model and solve problems involving continuous processes.
  • #1
kring_c14
76
0

Homework Statement



the repeating decimal . 27272727 . . . can be written as an infinite series. Write it as a series and tell if it diverges/converges. If it converges, find the sum.


Mickey, also a former student, knows how to do this one. Mickey knows enough math to write it as .27 + .0027 + .000027 . . .

It's now an infinite series. Mickey spots the common ratio of the series as .0027/.27 = .01. Therefore, it converges! He use the formula and presto:
S = .27/(1 - .01) = .27/.99 = 27/99 = 3/11
(remarkable achievement considering he's a mouse!)



The Attempt at a Solution



why did mickey wrote .0027 after .27?? why not .027??
 
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  • #2
because .27 + .0027 + .000027 = .272727
.27 + .027 = .297 -> .297 +.0027 = .2997 etc
 
  • #3
ahhh..haven't thought of it...thanks
 

FAQ: Infinite series (damn mickey mouse)

What is an infinite series?

An infinite series is a sum of an infinite number of terms. Each term is added to the previous one, resulting in an infinite sum.

What is the difference between a finite series and an infinite series?

A finite series has a limited number of terms, while an infinite series has an infinite number of terms.

What is the convergence of an infinite series?

The convergence of an infinite series refers to whether or not the series adds up to a finite number. If the series does add up to a finite number, it is said to be convergent. If the series does not add up to a finite number, it is said to be divergent.

How do you determine if an infinite series is convergent or divergent?

There are various tests that can be used to determine the convergence or divergence of an infinite series, such as the ratio test, the root test, and the integral test. These tests involve analyzing the behavior of the terms in the series as the number of terms approaches infinity.

What are some real-world applications of infinite series?

Infinite series are used in many fields, including physics, engineering, and finance, to model and solve problems involving continuous processes. For example, infinite series are used in calculus to calculate the area under a curve or the volume of a three-dimensional shape. They are also used in financial mathematics to model compound interest and in signal processing to analyze and manipulate continuous signals.

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