if f exist in the Riemann integral from [a,b] and if (P_n) is any sequence of tagged partitions of [a,b] such that the norm of these tagged partitions goes to 0 and lim_n S(g,P_n) exists.
To understand the question, we need to break it down into smaller parts.
1. Riemann Integral: The Riemann integral is a way to calculate the area under a curve. It is denoted by ∫ (pronounced as integral) and is used to find the accumulation of a function over an interval.
2. Tagged Partitions: A tagged partition of an interval [a, b] is a division of that interval into smaller subintervals, along with choosing a specific point within each subinterval called a tag. This is done in order to evaluate the function at that specific point.
3. Norm of a Tagged Partition: The norm of a tagged partition is defined as the length of the longest subinterval within that partition.
4. Sequence of Tagged Partitions: The sequence (Pn) refers to a collection of tagged partitions as n approaches infinity. In other words, it means we have an infinite number of smaller and smaller tagged partitions.
Now let's address the statement you mentioned:
"If f exists in the Riemann integral from [a, b] and if (Pn) is any sequence of tagged partitions of [a, b] such that the norm of these tagged partitions goes to 0 and limn S(g, Pn) exists."
Here's what this statement implies:
1. Function f exists in the Riemann integral from [a, b]: This means that the function f is integrable over the interval [a, b], and its Riemann integral exists.
2. Sequence of tagged partitions (Pn): We have a sequence of tagged partitions (Pn), which means we have an infinite collection of smaller and smaller tagged partitions of the interval [a, b].
3. Norm of tagged partitions goes to 0: The norm of these tagged partitions, which refers to the length of the longest subinterval in each partition, approaches 0 as n (the number of partitions) approaches infinity. This indicates that the tagged partitions become finer and finer.
4. Limit of S(g, Pn) exists: The limit as n approaches infinity of S(g, Pn) exists. Here, S(g, Pn) represents the Riemann sum for a given function g and tagged partition Pn. If the limit exists, it means that the Riemann sum converges to a specific value.
In summary, the statement says that if a function f is integrable over the interval [a, b], and we have a sequence of tagged partitions (Pn) such that the length of the longest subinterval in each partition approaches 0, and the Riemann sum for this sequence of tagged partitions converges to a limit, then we can conclude that the Riemann integral of f exists over the interval [a, b].