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Epsilon

$f(x + a\epsilon) = \sum_{n=0}^\infty f^{(n)}(x_0)a^n\epsilon^n/n!$.

$$ \begin{aligned} f(x + a\epsilon + b\epsilon^2) &= f(x + (a + b\epsilon)\epsilon) \\ &= \sum_{n=0}^\infty f^{(n)}(x)(a + b\epsilon)^n\epsilon^n/n! \\ &= \sum_{n=0}^\infty f^{(n)}(x) (\sum_{k=0}^n\binom{n}{k} a^{n-k} b^k\epsilon^k)\epsilon^n/n!\\ &= \sum_{k=0}^\infty \sum_{n=k}^\infty f^{(n)}(x) (\binom{n}{k} a^{n-k} b^k\epsilon^k)\epsilon^n/n!\\ &= \sum_{k=0}^\infty \sum_{n=0}^\infty f^{(n+k)}(x) (\binom{n+k}{k} a^n b^k\epsilon^k)\epsilon^{n + k}/(n + k)!\\ &= \sum_{k=0}^\infty b^k\epsilon^{2k} \sum_{n=0}^\infty f^{(n+k)}(x) (\binom{n+k}{k} a^n)\epsilon^n/(n + k)!\\ &= \sum_{k=0}^\infty b^k\epsilon^{2k}/k! \sum_{n=0}^\infty f^{(n+k)}(x) a^n\epsilon^n/n!\\ &= \sum_{k=0}^\infty b^k\epsilon^{2k}/k! d^k/dx^k \sum_{n=0}^\infty f^{(n)}(x) a^n\epsilon^n/n!\\ &= \sum_{k=0}^\infty b^k\epsilon^{2k}/k! d^k/dx^k f(x + a\epsilon)\\ \end{aligned} $$