I have a formula that looks a lot like integral: $$ m(t) = \lim_{\Delta t \to 0} \sum_{i \in \{\Delta t, 2\Delta t, ... ,t\} } ( m (i - \Delta t) + v(i)) \, \Delta t $$ where $v(t) = \text{const}$, $m(0)$ is given. Yet I wonder - how to turn it into an integral and is it possible when to calculate $m(t)$ I have to obtain $m\left(t- \Delta t\right)$ ?
2026-03-28 12:13:28.1774700008
How to turn recursion sum formula into an integral?
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The right side of your formula is a limit. If that limit exists, then it is an integral as you suggested. $\, m(t) = \int_0^t m(x) + v(x)\, dx. \,$ Your question about $\, m(t-\Delta t) \,$ turns out not to be a problem because in the limit that is the same as $\, m(t). \,$ Taking the derivative of both sides of the equation, $\, m'(t) = m(t) + v(t) \,$ which is a simple differential equation with initial value $\, m(0)=0. \,$