What Are the First Three Non-Zero Terms in the Series Expansion of (1-1/n)^1/n?

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The discussion focuses on finding the first three non-zero terms in the series expansion of (1-1/n)^1/n. The solution involves using the Taylor series expansion, specifically by substituting 1/n with x, which approaches zero. Participants suggest expanding (1+x)^(x) around x=0 and utilizing the exponential function and logarithm for simplification. The expected terms are identified as 1 - (1/n)^2 - 1/2(1-n)^3, with a request to find the term in (1/n)^4. The conversation emphasizes understanding the series expansion technique for accurate results.
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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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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)).
 
Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?

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