Exploring the Series Expansion of (1-1/n)^1/n in Ascending Powers of 1/n

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In summary, ascending powers in math refer to a number being raised to a progressively higher exponent. To solve them, you can use the rule a^m * a^n = a^(m+n). The pattern in ascending powers is that the base number stays the same and the exponent increases by 1 each time. The main difference between ascending and descending powers is the direction of the exponent. Ascending powers are commonly used in real life in fields like finance, science, and population growth.
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Homework Statement


Show that the first three non-zero terms in the series expansion of (1-1/n)^1/n in ascending powers of 1/n are 1-(1/n)^2-1/2(1-n)^3 and find the term in (1/n)^4


Homework Equations



Macclaurin? Taylor?

The Attempt at a Solution



Can someone please point me on where I should touch this problem? I don't think I understand the problem correctly.
 
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  • #2


Taylor series. Take 1/n=x to be a number near zero. Expand (1+x)^(x) around x=0. Hint: that's exp(x*log(1+x)).
 

FAQ: Exploring the Series Expansion of (1-1/n)^1/n in Ascending Powers of 1/n

What are ascending powers in math?

Ascending powers in math refer to a mathematical concept where a number is raised to a progressively higher exponent. For example, ascending powers of 2 would be 21, 22, 23, and so on.

How do you solve ascending powers?

To solve ascending powers, you can use the rule that states am * an = am+n. This means that if you have an ascending power like 23, you can break it down to 22 * 21, which equals 4 * 2 = 8.

What is the pattern in ascending powers?

The pattern in ascending powers is that the base number remains the same, and the exponent increases by 1 each time. For example, in ascending powers of 3, the pattern would be 31, 32, 33, and so on.

What is the difference between ascending and descending powers?

The main difference between ascending and descending powers is the direction in which the exponent increases or decreases. Ascending powers have a progressively higher exponent, while descending powers have a progressively lower exponent.

How are ascending powers used in real life?

Ascending powers are used in various applications, such as in computing exponential growth and decay in finance or population growth. They are also used in scientific fields like physics and chemistry to calculate exponential functions and rates of change.

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