Summation of 1.05^n/n^5: Help Needed!

  • Thread starter lmannoia
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In summary, the formula for "Summation of 1.05^n/n^5" is: ∑(1.05^n/n^5) = 1 + 1.05/1^5 + 1.05^2/2^5 + 1.05^3/3^5 + ... + 1.05^n/n^5. To calculate this formula, you plug in the value of n and add all the terms together. The pattern in this formula can be seen in the powers of 1.05 and n, with a decreasing pattern in the terms of the summation. As n approaches infinity, the limit of the formula is 1. This formula can be used in
  • #1
lmannoia
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Homework Statement



Summation from 1 to infinity of 1.05^n/n^5

Homework Equations





The Attempt at a Solution


Lost. I'm not sure if the ratio test would apply here.. convergence tests are definitely not my strong point!
 
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  • #2
Does the limit of [tex]\frac{1,05^n}{n^5}[/tex] approach 0? What can you conclude from this?
 
  • #3
No, it would approach infinity, right? Meaning that it diverges?
 
  • #4
Yes, so you can conclude that the series diverges.
 

1. What is the formula for "Summation of 1.05^n/n^5"?

The formula for "Summation of 1.05^n/n^5" is:
∑(1.05^n/n^5) = 1 + 1.05/1^5 + 1.05^2/2^5 + 1.05^3/3^5 + ... + 1.05^n/n^5

2. How do you calculate "Summation of 1.05^n/n^5"?

To calculate "Summation of 1.05^n/n^5", you need to plug in the value of n into the formula and then add all the terms together. For example, if n=3, the calculation would be:
∑(1.05^n/n^5) = 1 + 1.05/1^5 + 1.05^2/2^5 + 1.05^3/3^5 = 1 + 1.05/1 + 1.05^2/32 + 1.05^3/243 = 1 + 1.05 + 0.033 + 0.004 = 2.087

3. What is the pattern in "Summation of 1.05^n/n^5"?

The pattern in "Summation of 1.05^n/n^5" can be seen in the powers of 1.05 and the powers of n. As n increases, the powers of 1.05 also increase, but at a slower rate. This leads to a decreasing pattern in the terms of the summation.

4. What is the limit of "Summation of 1.05^n/n^5" as n approaches infinity?

The limit of "Summation of 1.05^n/n^5" as n approaches infinity is 1. This means that as n gets larger and larger, the sum of all the terms in the series will approach 1 as the terms become smaller and smaller.

5. Can "Summation of 1.05^n/n^5" be used in real-world applications?

Yes, "Summation of 1.05^n/n^5" can be used in various real-world applications such as finance, population growth, and physics. The formula can model the compound interest in investments, the population growth of a species, and the distance traveled by an object under constant acceleration, among others.

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