Finding a definite integral from the Riemann sum

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The discussion focuses on finding a definite integral using a Riemann sum with the interval from 1 to 2. The user initially struggles with the logarithmic expression derived from the sum but later simplifies it. By rewriting the argument of the logarithm, they conclude that the function can be expressed as f(x) = xlnx. This simplification helps clarify the approach to solving the integral. The conversation highlights the importance of simplifying expressions in calculus problems.
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Homework Statement
Consider the following limit of a Riemann sum for a function f on [a, b]. Identify f, a, and b,
and express the limit as a definite integral.

*see actual expression in the description below. it was too complicated to type out so I included a picture instead.
Relevant Equations
∆x = (b-a)/n
xiR = a + i∆x
1679251462458.png


Hi! I am having trouble finalizing this problem.

The interval is given so we know that a = 1 and b = 2. From there you can figure out that ∆x = 1/n, xiR = 1 + i/n.
Using logarithmic properties, I rearranged the expression and wrote (1 + i/n)(1/n)ln[(n + i)/n].
I can guess that the function is going to look something like this: f(x) = xln(...) but I don't know what goes in the logarithmic function...

I have been trying to rewrite it in terms of xiR but no luck :(
any help would be really appreciated. thank you !
 
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Can you simplify the argument of the logarithm any further?
 
pasmith said:
Can you simplify the argument of the logarithm any further?
Yes, I realized I was overcomplicating things. If I simplify (n+i)/n to (n/n +i/n) to (1+ i/n), I get f(x) = xlnx
 

Thread 'Does this series converge uniformly?'

First, I tried to show that ##f_n## converges uniformly on ##[0,2\pi]##, which is true since ##f_n \rightarrow 0## for ##n \rightarrow \infty## and ##\sigma_n=\mathrm{sup}\left| \frac{\sin\left(\frac{n^2}{n+\frac 15}x\right)}{n^{x^2-3x+3}} \right| \leq \frac{1}{|n^{x^2-3x+3}|} \leq \frac{1}{n^{\frac 34}}\rightarrow 0##. I can't use neither Leibnitz's test nor Abel's test. For Dirichlet's test I would need to show, that ##\sin\left(\frac{n^2}{n+\frac 15}x \right)## has partialy bounded sums...
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