What Is the Correct Sum of the Series \( \sum_{n=1}^\infty \frac{8^n}{9^n} \)?

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In summary, the sum of the series \sum_{n=1}^\infty \frac{2^n+6^n}{9^n} can be written as two separate geometric series: \sum_{n=1}^\infty (\frac{2}{9})^n and \sum_{n=1}^\infty (\frac{6}{9})^n. Using the geometric series formula, the sum of each series can be calculated, resulting in a final answer of 30/7.
  • #1
ILoveBaseball
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Determine the sum of the following series

[tex]\sum_{n=1}^\infty \frac{2^n+6^n}{9^n}[/tex] or can be written as...
[tex]\sum_{n=1}^\infty \frac{8^n}{9^n}[/tex]

[tex] A_1 = 8/9, A_2 = 64/81, A_3 = 512/729 [/tex]

common ration (r)= 8/9
first term (a)= 8/9

so plugging everything i know into the geometric series formula:
[tex] \frac {a}{1-r}[/tex]

i get... 8
but it's wrong, and i don't see why
 
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  • #2
[tex]\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}[/tex]

And you know what:

[tex]\frac{k^n}{m^n}[/tex]

is with k and m constants right?
 
Last edited:
  • #3
Are you sure 2^n + 6^n = 8^n?
 
  • #4
saltydog said:
[tex]\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}[/tex]

And you know what:

[tex]\frac{k^n}{m^n}[/tex]

is with k and m constants right?



[tex]\frac{2}{9}\sum_{n=1}^\infty \frac{1^n}{1^n}[/tex] +

[tex]\frac{6}{9}\sum_{n=1}^\infty \frac{1^n}{1^n}[/tex]


right?
 
  • #5
I strongly suggest you go back and review the section on exponents in your earlier textbooks -- your problem lies with the fact you don't know how to manipulate them, and it would be much easier to refresh your memory without having to worry about calculus stuff at the same time!
 
  • #6
Testing "2" in for x yields the following statements.

[tex]2^2 + 6^2 = 40[/tex]
[tex]8^2 = 64[/tex]

Hurkyl is right. Rework that step and see what else you can come up with.
 
  • #7
i got 4. final answer regis :tongue2:
(saltydog did 80% of the work)
 
  • #8
the answer is 2.2857, it was actually an easy question. thanks for the help
 
  • #9
Really? I find it strange that it happened to be a terminating decimal.
 
  • #10
ILoveBaseball said:
the answer is 2.2857, it was actually an easy question. thanks for the help

how did you get that? i did [tex]\frac{1}{1- 2/9} + \frac{1}{1-2/3}[/tex] which adds up to 4
 
  • #11
Check the starting index of the sum.
 
  • #12
[tex]\sum_{n=1}^\infty \frac{2^n+6^n}{9^n}[/tex]

So, that's what we're dealing with, right? I'm not too sharp on series, but I don't thing this is hard.

Follow saltydog's advice, and the series will become two series,

[tex]\sum_{n=1}^\infty \frac{2^n}{9^n} + \sum_{n=1}^\infty \frac{6^n}{9^n}[/tex]

[tex]\sum_{n=1}^\infty (\frac{2}{9})^n + \sum_{n=1}^\infty (\frac{6}{9})^n[/tex]

Use the geometric series formula on each of those.

fourier jr said:
i did [tex]\frac{1}{1- 2/9} + \frac{1}{1-2/3}[/tex] which adds up to 4

It does?
 
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  • #13
sorry 30/7, not 4
 

What is the formula for the sum of the series 8^n/9^n?

The formula for finding the sum of the series 8^n/9^n is S = a/(1-r), where a is the first term (in this case, 8/9) and r is the common ratio (in this case, 8/9).

What is the limit of the sum of the series 8^n/9^n as n approaches infinity?

The limit of the sum of the series 8^n/9^n as n approaches infinity is 1. This means that as n gets larger and larger, the sum of the series gets closer and closer to 1.

How many terms are needed to approximate the sum of the series 8^n/9^n within a certain margin of error?

The number of terms needed to approximate the sum of the series 8^n/9^n within a certain margin of error depends on the value of the margin of error and the desired level of accuracy. Generally, the more terms that are included in the sum, the more accurate the approximation will be.

What is the relationship between the sum of the series 8^n/9^n and the geometric series 8^n?

The sum of the series 8^n/9^n is a specific case of the geometric series 8^n, where the common ratio is less than 1. The sum of the series 8^n/9^n is equal to the first term (8/9) divided by 1 minus the common ratio (8/9), which is the same as the formula for the sum of a geometric series with a common ratio less than 1.

What real-world applications use the concept of the sum of the series 8^n/9^n?

The concept of the sum of the series 8^n/9^n is used in various fields such as finance, engineering, and computer science. For example, in finance, it can be used to calculate the future value of an investment with a constant interest rate. In engineering, it can be used to model exponential growth or decay. In computer science, it can be used to analyze algorithms with a constant growth rate.

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