Question

Investigate the following problem for some small values of n, and then explain the general statement based on the binomial theorem.
What is the coefficient of h (i.e. h^1) in the expansion of (1+h)^n-1/(h)?

Answer 1

\[h^{n-2}\]

Answer 2

can you please elaborate a little more. This problem really confusing to me for some reason

Answer 3

is it \[(1+h)^{\frac{ n-1 }{ h }}\] or\[\frac{ (1+h)^{n-1} }{ h }\]

Answer 4

They mean:

Numerator: Expand \((1+h)^n\) and subtract \(1\) from it. Turns out that the constant disappears, so, the terms always contains at least one factor \(h\). For that reason, it is possible to divide it by \(h\) and still have a polynomial:

Expand: \((1+h)^n = 1 + nh + \binom{n}2 h^2 + \binom{n}3 h^3 + \dotsb + h^n\)

Remove "1": \((1+h)^n - 1 = nh + \binom{n}2h^2 + \dotsb + h^n\)

Divide: \(\frac{(1+h)^n-1}{h} = \frac{nh + \binom{n}2h^2 + \dotsb + h^n}h = n+\binom{n}2 h + \dotsb + h^{n-1}\).

What is the coefficient of \(h^1\) ?

Answer 5

*I think they mean*