Sequence of 3/4^(2k). Show is convergent, find sum

In summary, the sequence 3/4^(2k) is a mathematical sequence where each term is obtained by raising 3/4 to the power of 2k. To show that the sequence is convergent, we can use the limit comparison test or show that it is a geometric series with a common ratio less than 1. The sum of the sequence can be calculated using the formula for the sum of a geometric series, and the sequence can be simplified to (3/16)^k. This sequence has various real-life applications, such as modeling exponential decay processes and financial calculations.
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Homework Statement



Consider the sequence ak = (3)/(4^(2k)). Show is convergent, find sum. Please check work.

Homework Equations



ak = 3/(4^2k)

let s {n} be the series associated with the sequence. Cannot write summation notation here, but k starts at 1 (k = 1 on bottom) and infinity on top.

The Attempt at a Solution



ak1 = 0.1875
ak2 = 0.01171875
ak3 = 0.000732421876
ak4 = 0.00004577636719

ak4/ak3 = 0.0625

Convergence: 1) r = 0.0625. r < 1, ∴ convergent
Sum: 2) (0.1875)/(1-0.0625) = 0.2. Sum is 0.2
 
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It is correct, if the summation starts with k=1.

ehild
 
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ehild said:
It is correct, if the summation starts with k=1.

ehild

Thanks! Yes, it does.
 

What is the sequence 3/4^(2k)?

The sequence 3/4^(2k) is a mathematical sequence where each term is obtained by raising 3/4 to the power of 2k. This means that the first term is 3/4^0, the second term is 3/4^2, the third term is 3/4^4, and so on.

How do you show that the sequence is convergent?

In order to show that the sequence 3/4^(2k) is convergent, we need to prove that the terms of the sequence approach a finite limit as k approaches infinity. This can be done by using the limit comparison test or by showing that the sequence is a geometric series with a common ratio less than 1.

What is the sum of the sequence 3/4^(2k)?

The sum of the sequence 3/4^(2k) can be calculated by using the formula for the sum of a geometric series: S = a/(1-r), where a is the first term and r is the common ratio. In this case, a = 3/4^0 = 1 and r = 3/4^2 = 9/16. Therefore, the sum is S = 1/(1-9/16) = 16/7.

Can the sequence be written in a simplified form?

Yes, the sequence 3/4^(2k) can be written in a simplified form as (3/16)^k. This is because 3/4^2 = 3/16, and raising this value to the power of 2k is equivalent to multiplying 3/16 by itself k times.

What are some real-life applications of this sequence?

This sequence can be used to model exponential decay processes, such as radioactive decay or bacterial growth. It can also be applied in financial calculations, such as compound interest or amortization of loans.

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