Infinite Geometric Series and Convergence

In summary, for part a, the common ratio is $\frac{1}{4}$. For part b, the infinite series is a geometric series. For part c, the exact value of the common ratio is $\frac{\sqrt{5}}{2}$. And for part d, the series diverges as the common ratio is greater than 1.
  • #1
karush
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a. Find the common ration $r$, for an infinite series
with an initial term $4$ that converges to a sum of $\displaystyle\frac{16}{3}$
$$\displaystyle S=\frac{a}{1-r} $$ so $\displaystyle\frac{16}{3}=\frac{4}{1-r}$ then $\displaystyle r=\frac{1}{4}$

b. Consider the infinite geometric series

$\displaystyle
\frac{8}{25} + \frac{4\sqrt{5}}{25}+\frac{2}{5}+\frac{\sqrt{5}}{5}+\cdot\cdot\cdot$

c. What is the exact value of the common ratio of the series
$\frac{8}{25}\cdot r = \frac{4\sqrt{5}}{25}$ so $r=\frac{\sqrt{5}}{2}$d. Does the series converge?

not sure how to do this?
 
Last edited:
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  • #2
Hi karush,

Your final answer for part a is correct.
To find the common ratio, $r$, solve $\dfrac{16}{3}=\dfrac{4}{1-r}$ for $r$.
Please show your work.

Part d:

Is $\dfrac{\sqrt5}{2}$ greater than, less than or equal to $1$? What does your answer to this question imply?
 
  • #3
a. Find the common ration $r$, for an infinite series
with an initial term $4$ that converges to a sum of $\displaystyle\frac{16}{3}$
$$\displaystyle S=\frac{a}{1-r} =$$
so $\displaystyle\frac{16}{3}=\frac{4}{1-r} \\
3\cdot4 = 16\left(1-r\right)\implies12=16-16r\implies4=16r\implies r=\frac{4}{16}=\frac{1}{4}$b. Consider the infinite geometric series

$\displaystyle
\frac{8}{25} + \frac{4\sqrt{5}}{25}+\frac{2}{5}+\frac{\sqrt{5}}{5}+\cdot\cdot\cdot$

c. What is the exact value of the common ratio of the series
$\frac{8}{25}\cdot r = \frac{4\sqrt{5}}{25}$ so $r=\frac{\sqrt{5}}{2}$d. Does the series converge?

$\frac{\sqrt{5}}{2}=1.11$ $r\ge1$ no it divergess
 
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  • #4
$r>1$

Correct. :)
 

FAQ: Infinite Geometric Series and Convergence

What is an infinite series?

An infinite series is a mathematical concept in which an infinite number of terms are added together to form a sum. Each term in the series is related to the previous one by a specific pattern or rule.

How do you find the sum of an infinite series?

The sum of an infinite series can be found by adding together all the terms in the series. However, since the series has an infinite number of terms, it is impossible to add them all together. Instead, mathematicians use various techniques and formulas, such as the geometric series formula or the telescoping series method, to find the sum.

What is the difference between a convergent and a divergent infinite series?

A convergent series is one in which the sum of all the terms in the series approaches a finite number as the number of terms increases. On the other hand, a divergent series is one in which the sum of the terms either approaches infinity or does not approach a specific value at all.

What is the significance of infinite series in mathematics?

Infinite series are important in many areas of mathematics, including calculus, number theory, and physics. They allow us to approximate values, calculate areas and volumes, and solve complex equations. Infinite series also have applications in finance, economics, and computer science.

How can infinite series be used to solve real-world problems?

Infinite series can be used to model real-world phenomena, such as population growth, interest rates, and radioactive decay. By understanding the patterns and rules of infinite series, we can make predictions and solve problems in fields such as finance, engineering, and biology.

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