- #1
lmannoia
- 32
- 0
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!
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 "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
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.
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.
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.