Determine the maximal addend in Newton Binomial Expansion of the expression $$\left ( 2n+\frac{1}{2n} \right )^{4n+1},\quad \left ( \forall n \in \mathbb{N} \setminus \left \{ 1 \right \} \right )$$
Expanded expression looks like this:
$$\left ( 2n+\frac{1}{2n} \right )^{4n+1}=\sum_{k=0}^{4n+1}{\binom{4n+1}{k} \left ( 2n \right )^{4n+1-k}\left (\frac{1}{2n} \right )^k}=\sum_{k=0}^{4n+1}{\binom{4n+1}{k}\left( 2n\right)^{4n+1-2k}}$$
What I am asking is: Satisfying what condition would provide maximum addend?
Also, how does the domain $n \in \mathbb{N} \setminus \left \{ 1 \right \}$ affect this expansion?
2026-04-02 20:08:22.1775160502
Determine maximal addend in Newton Binomial Expansion.
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