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Definite integral is the limit of Riemann sum.

$\lim\limits_{x \to a} f(x)=L$ is defined as:

For every $\varepsilon>0$, we can find a $\delta>0$ such that $\left| {f\left( x \right) - L} \right| < \varepsilon \hspace{0.5in}{\mbox{whenever}}\hspace{0.5in}0 < \left| {x - a} \right| < \delta$

In an analogous way (or in a different way), how can we define the limit of a sum?

bof
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Joe
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  • Limit of Riemann sum is somewhat different from limit of functions (the one you mention in your question). There are two approaches here which are discussed in https://math.stackexchange.com/a/2047959/72031 – Paramanand Singh Jun 22 '19 at 16:34

1 Answers1

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The formal definition of an integral is that $\int_a^b f(x)dx=I$ if for every $\epsilon>0$ we can find $\delta>0$ such that for any partition $a=x_0<x_1<...<x_n=b$ of $[a,b]$ which satisfies $\max (x_i-x_{i-1})<\delta$ and for any choice of points $\{t_i\}_{i=1}^n, t_i\in [x_{i-1},x_i]$ we have that the distance of the corresponding Riemann sum from $I$ is at most $\epsilon$. So it is more than just a limit, it must not depend on partition and choice of points.

Mark
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