Rn -5 +i n ||
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Equations and inequalities describe the relationship between two mathematical expressions.
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A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
![**Express the limit of \( R_n \) as a definite integral, identifying the correct interval.**
\[
R_n = \frac{2}{n} \sum_{i=1}^{n} \left( -5 + i \frac{2}{n} \right).
\]
\[
\lim_{n \to \infty} R_n = \int_{[ \quad ]}^{[ \quad ]} \left[ \quad \right] \, dx.
\]
The problem involves expressing the limit of a Riemann sum \( R_n \) as a definite integral and identifying the interval of integration. The equation provided for \( R_n \) suggests a partitioning of an interval into \( n \) subintervals, each of width \( \frac{2}{n} \), and evaluating the sum of function values multiplied by this width. The terms in the sum \(-5 + i \frac{2}{n}\) adjust according to \( i \), indicating the \( x \)-values being sampled as \( n \to \infty \).
The right side of the equation shows an incomplete integral, suggesting that the reader should complete it by identifying the limits of integration and the function to integrate based on the given Riemann sum structure.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5b46b3ce-7e6b-4c27-9355-f7cbdeacdab7%2Fd719f0e1-ecfc-46fb-96da-14902bf6c085%2Fb9i6b8a_processed.png&w=3840&q=75)
Transcribed Image Text:**Express the limit of \( R_n \) as a definite integral, identifying the correct interval.**
\[
R_n = \frac{2}{n} \sum_{i=1}^{n} \left( -5 + i \frac{2}{n} \right).
\]
\[
\lim_{n \to \infty} R_n = \int_{[ \quad ]}^{[ \quad ]} \left[ \quad \right] \, dx.
\]
The problem involves expressing the limit of a Riemann sum \( R_n \) as a definite integral and identifying the interval of integration. The equation provided for \( R_n \) suggests a partitioning of an interval into \( n \) subintervals, each of width \( \frac{2}{n} \), and evaluating the sum of function values multiplied by this width. The terms in the sum \(-5 + i \frac{2}{n}\) adjust according to \( i \), indicating the \( x \)-values being sampled as \( n \to \infty \).
The right side of the equation shows an incomplete integral, suggesting that the reader should complete it by identifying the limits of integration and the function to integrate based on the given Riemann sum structure.
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