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

Question

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

turingtest · Answer

are you familiar with binomial coefficients?

turingtest · Answer

$$(a+b)^n=\sum_{k=0}^n\binom n ka^kb^{n-k}$$

anonymous · Answer

ive tried expanding it but still got the 3rd term wrong :/

turingtest · Answer

you don't have to expand the whole thing to find a particular term. k goes from 0 to n, so there are n+1 terms total. if you want term t, then (since you start counting from zero) you simply plug in t-1 for k in the formula above without the summation sign. for the first term, plug in k=0, for the second, plug in k=1, etc.