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\displaystyle\sum_{n=2}^{\infty}a_{n-1}z^n \stackrel{(1)}{=}z\displaystyle\sum_{n=2}^{\infty}a_{n-1}z^{n-1} \stackrel{(2)}{=}
z\displaystyle\sum_{n=1}^{\infty}a_{n}z^n \stackrel{(3)}{=}
z\biggr( \underbrace{ z\displaystyle\sum_{n=1}^{\infty}a_{n}z^n+a_0}_{G(z)} - a_0\biggr)\stackrel{(4)}{=}zG(z).
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