For positive integers $m$, $n$, determine the coefficient of $x^0$ in \( \left( x + \frac{1}{x

Question

For positive integers $m$, $n$, determine the coefficient of $x^0$ in $ \left( x + \frac{1}{x^m} \right)^n$ Hence determine all pairs $ (m, n )$ for which a constant term exists.

Accepted Answer

Coefficient $\binom{n}{\frac n{m+1}}$ if $m+1$ divides $n$; $0$ otherwise. All pairs: $(m,n)=(m,(m+1)k)$, for positive integers $m,k$.

Binomial Theorem - General Terms and Coefficients in Binomial Expansions

Binomial Theorem - General Terms and Coefficients in Binomial Expansions

Solution:

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