<s>[INST] Prove that $\left (1+\dfrac{1}{2n}\right )^n-\left (1-\dfrac{1}{2n}\right)^n\geq 1$. Let $n\in\Bbb N$, then $\left (1+\dfrac{1}{2n}\right )^n-\left (1-\dfrac{1}{2n}\right)^n\geq 1.$
Notice that,
$$\begin{align*}\left (1+\dfrac 1 {2n}\right )^n-\left (1-\dfrac{1}{2n}\right)^n&=\sum_{k=0}^n \binom{n}{k} \left (\dfrac{1}{2n}\right )^k-\sum_{k=0}^n\binom{n}{k}\left (-\dfrac{1}{2n}\right )^k\\
&=\sum_{k=0}^n \binom{n}{k}\left (\dfrac{1}{2n}\right )^k[1-(-1)^k] \end{align*}$$
I don't know how to get from there, because if $ n $ falls into an even number, all that goes to 0, and it is not greater than or equal to 1, but if $ n $ is odd if it is true. How do I finish the problem?
 [/INST] At least for large values of $n$
$$a_n=\left (1+\dfrac{1}{2n}\right )^n-\left (1-\dfrac{1}{2n}\right)^n$$
$$b=\left (1+\dfrac{1}{2n}\right )^n\implies \log(b)=n \log\left (1+\dfrac{1}{2n}\right )=\frac{1}{2}-\frac{1}{8 n}+\frac{1}{24 n^2}+O\left(\frac{1}{n^3}\right)$$
$$b=e^{\log(b)}=\sqrt e \left(1-\frac{1}{8 n}+\frac{19}{384 n^2}+O\left(\frac{1}{n^3}\right)\right)$$
$$c=\left (1-\dfrac{1}{2n}\right )^n\implies \log(c)=n \log\left (1-\dfrac{1}{2n}\right )=-\frac{1}{2}-\frac{1}{8 n}-\frac{1}{24 n^2}+O\left(\frac{1}{n^3}\right)$$
$$c=e^{\log(c)}=\frac{1}{\sqrt{e}} \left(1-\frac{1}{8 n}-\frac{13}{384 n^2}+O\left(\frac{1}{n^3}\right) \right )$$
$$a_n=2 \sinh \left(\frac{1}{2}\right)-\frac{\sinh \left(\frac{1}{2}\right)}{4 n}+\frac{13+19
   e}{384 \sqrt{e} n^2}+O\left(\frac{1}{n^3}\right)$$
$$2 \sinh \left(\frac{1}{2}\right)-\frac{\sinh \left(\frac{1}{2}\right)}{4 n}\gt 1 \implies n > \frac{1}{8-4 \text{csch}\left(\frac{1}{2}\right)}\sim 3.08774$$
Computed exactly
$$a_{10}=\frac{26372036801}{25600000000}=1.030158$$ while the truncated series gives
$$a_{10}=\frac{37939 e-37907}{38400 \sqrt{e}}\sim 1.030184$$
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