Question

whats the equation for the sum of the series 1/(n^2+i^2)? (this is a riemann sum question, so i'll take the limit as n approaches infinity later; i guess, for now, we're assuming n is constant?)

Answer 1

turning test! could you look at the question i posted and see if you know by chance :)

Answer 2

you are computing

\[\lim_{n\to\infty}\sum_{i=1}^{n}\frac{1}{n^2+i^2}\]?

Answer 3

Simple, you need the formal definition of the integral:
\[ \int_{0}^{1} f(x) \;dx= \lim_{n \rightarrow\infty} \frac{1}{n}\sum_{i=1}^{n} f(\frac{i}{n}) \]

Note: It is not straightfoward to understand why the limits of integration are "0" and "1", think about them and if you don't get it just ask and I will explain.

Now, all we need to do is get the summation you are given to look like the right hand side, if we can achieve this then we are allowed to state it as the integral on the left hand side. Making the evaluation alot easier than if we had the summation instead. So, here we go:

\[ \lim_{n \rightarrow\infty} \sum_{i=1}^{n} \frac{1}{n^{2}+i^2}\] \[\lim_{n \rightarrow\infty} \sum_{i=1}^{n} \frac{1}{n^2} \frac{1}{1+(\frac{i}{n})^2}\]
\[\lim_{n \rightarrow\infty} \frac{1}{n^2} \sum_{i=1}^{n} \frac{1}{1+(\frac{i}{n})^2} \]
\[\lim_{n \rightarrow\infty} \frac{1}{n} \;\; \frac{1}{n} \sum_{i=1}^{n} \frac{1}{1+(\frac{i}{n})^2} = \lim_{n \rightarrow\infty} \frac{1}{n} \int_{0}^{1} \frac{1}{1+x^2}\]
So,
\[ \lim_{n \rightarrow\infty} \frac{1}{n} \;\;\ Tan^{-1}(x)|_{0}^{1}= \lim_{n \rightarrow\infty} \frac{1}{n} (\frac{\pi}{ 4}-0)=0\]

And Bingo!

Answer 4

yes...very easy if that is what they wanted

Answer 5

To answer your question: Yes, in some sense we do assume "n" to be a constant. Just that it is a very large constant, the bigger the more accurate the summation. As it goes to infinity it becomes the integral. Remember "n" is just the number of rectangles you are using to approximate the area under the function.