Sums Definition and 349 Threads

I have a homework question that reads: Represent the following rational functions as sums of elementary fractions and find the primitive functions ( indefinite integrals ); (a) f(z)=z-2/z^2+1 But my confusion arrises when I read sums of elementary fractions. I think what the question is...

So I'm taking an introductory number theory course as an undergraduate, and this particular "genre" of questions really just has me stumped. Pick a prime p such that p is odd. Now, take various sums up of natural numbers from 1 to p, and show that the results are divisible by p. For...

hi i m hashim i want to solve a qquestion 1.if x is rational & y is irrational proof x+y is irrational? 2. if x not equal to zero...y irrational proof x\y is irrational?? 3.if x,y is irrational ..dose it implise to x+y is irrational or x*y is irrational thanks please hashim

I was trying to build a probability-related software package and needed to have a theoretical framework to deal with some less common issues (i.e. stuff that you don't find in the average textbook). I was hoping that somebody could give me pointers as to where to find the proper formulae...

Cantor set defined via sums, whaaaaa?!? problem 19 chapter 3 of Rudin. I'm totally lost, I've even done a project on the Cantor set before but I just don't know where to start here. Associate to each sequence a=(p_n) in which p_n is either 0 or 2, the real number x(a) = sum from 1 to...

Howdy everyone. I'm not very good at writing proofs, so I am wondering if someone can tell me if I'm even on the right page with this. I am not sure if I understand the idea correctly. The theorem goes as follows: Suppose V is a finite dim. vector space with subspaces U and W such that V is the...

Homework Statement The following sum \sqrt{9 - \left(\frac{3}{n}\right)^2} \cdot \frac{3}{n} + \sqrt{9 - \left(\frac{6}{n}\right)^2} \cdot \frac{3}{n} + \ldots + \sqrt{9 - \left(\frac{3 n}{n}\right)^2} \cdot \frac{3}{n} is a right Riemann sum for the definite integral. Solve as n->infinity...

Homework Statement The sum of two numbers is 20. What is the least possible sum of their squares. 2. The attempt at a solution Before I show my work, I'm pretty sure I have the answer. I think it's 200. If you add 10 and 10, you will have 20. If you square 10 you get 100, thus the sum...

Hi guys, I'm working on a project at the moment where I want to convert a sum to an integral but I am out of ideas. Basically I have something like: Sum over h: [f(h+0.5dh)]^(-1) - [f(h-0.5dh)]^(-1) where h goes from 0 to H. Any tips would be appreciated!

Use the riemann sums model to estimate the area under the curve f(x) = x^2, between x =2 and x = 10, using an infinite number of strips. Be sure to include appropriate diagriams and full explanation of the method of obtaining all numerical values, full working and justification. Does anybody...

Here is the question from the book: ------ Let n\geq1 and let a_1,...,a_n and b_1,...,b_n be real numbers. Verify the identity: \left(\sum_{i=1}^n{a_ib_i}\right)^2 + \frac{1}{2}\sum_{i=1}^n{\sum_{j=1}^n{\left(a_ib_j-a_jb_i\right)^2}} =...

I do have a series of channels that contain the number of radioactive counts within a small energy range. Since the occurence of radioactive decay is statistical, the error in the number of counts is simply the square of the number of counts. Each channel contains counts from two different...

Homework Statement The voltage across a resistor is given by: v(t) = 5 + 3 \cos{(t + 10^o)} + \cos{(2 t + 30^o)} V Find the RMS value of the voltage Homework Equations For a periodic function, f(t), the rms value is given by: f_{rms} (t) = \sqrt{\frac{1}{T} \int_{0}^{T} f(t)^2 dt} Where T...

I don't think I *really* understand the Central Limit Theorem. Suppose we have a set of n independent random variables \{X_i\} with the same distribution function, same finite mean, and same finite variance. Suppose we form the sum S_n = \sum_{i=1}^n X_i. Suppose I want to know the...

Is there ever a situation where it is more appropriate/advantageous to use Riemann summation as opposed to evaluating an integral, or is Riemann summation merely taught in order to help the student to understand what's going on?

Hello everyone I'm studying for my next exam and I screwed up on the geometric progressions and arthm and they are the easiest of them all but I don't know what I'm doing wrong. The first problem on the exam said: Suppose that an arithmetic series has 202 terms. If the first term is 4PI and...

Hello everyone, I was wondering if someone can double check my work for #'s 25 and 31. The directions say: Express each of the sums in closed form (without using a summation symbol and without using an ellipis...).http://img224.imageshack.us/img224/3514/lastscanle2.jpg Thanks!

I have just made the following variable switch: \sum_{i=0}^n\sum_{j=0}^m\binom{n}{i}\binom{m}{ j}x^{i+j}=\sum_{k=0}^{n+m}\sum_{i=0}^k\binom{n}{i}\binom{m}{k-i}x^{k} I know it's right, but is there a method I can use to prove without a shadow of a doubt that it is?

Let A(n) = {1,2,3,4, … 8} and B(n) = {1,2,3,4, …. 16} be two sets of consecutive integers with no repetition. Divide the set of elements B(n) into two subsets C(n) and D(n) each comprising 8 different integers and such that every element of B(n) is used with the condition that 8 equations...

Can someone confirm/disprove the following: If X\subset\mathbb{P} is infinite, then \sum_{n\in X}\frac{1}{n} diverges or is irrational.

Some sums, don't sum up :) I have a problem that require some math tricks, and after I tried to solve it myself I looked at the answer and I don't understand how this is done : \[ \sum\limits_{k = 0}^n {\left( {\frac{2}{5}} \right)^k } + \sum\limits_{k = 0}^n {\left( {\frac{3}{5}}...

Which squares are expressible as the sum of two squares? Is there a simple expression I can write down that will give me all of them? Some of them? Parametrization of the pythagorean triples doesn't seem to help.

1) develop the function f(x)=(e^x)sin(x) into a power sum over the point 0. 2) find the convergence radius R of \sum_{\substack{0<=n<\infty}}\frac{(n!)^2}{(2n)!}x^n and say if it converges or diverges at x=-R, x=R. about the second question i got that R=4, through hadamard test, but i didnt...

I have a problem I'm working on where the general premise is that there is a box being pushed along the ceiling at a constant speed. The force F is at some angle with respect to the vertical. There is a coefficient of kinetic friction between the box and the ceiling and the persons hands and the...

Which command in Mathematica will give me the sum of the digits of a number in base 10? Thanks.

"strange" sums... Let be the next 2 sums in the form: f(x)+f(x+1)+f(x+2)+... (1) and f(x)+f(2x)+f(3x)+... (2) how would you calculate them?..well i used a "non-rigorous" but i think correct method to derive their sums..let be the infinitesimal generator D=d/dx...

I am doing exactly what this article is saying in order to try to graph a sequence of partial sums. If I enter the syntax, given below, on the home screen, it will list the first 25 terms of the partial sum. How do I graph those terms using sequence mode...

Hi, I just need a little help in getting some sums. Can anyone of you give me a site where I can find sums in Log so that I can do them and practice a lot. I mean like sums in this type Show that log(xy)base16 = 1/2log(X)base4 + 1/2log(Y)base4 Thanks just need some sums of this type...

I need to calculate \sum_{i=1}^n \frac{1}{i(i+1)} useing the fact that: \sum_{i=1}^n F(i) - F(i-1) = F(n) - F(0) now I chose the function F(i) = \frac{1}{i} \frac{1}{(i+1)} ... \frac{1}{(i+r)} so F(i)-F(i-1)=(\frac{1}{i}\frac{1}{(i+1)} ...

This discussion is that of converting infinite series to infinite products and vice-versa in hopes of, say, ending the shortage of infinite product tables. Suppose the given series is \sum_{k=0}^{\infty} a_k Let S_n[/tex] denote the nth partial sum, viz. S_n=\sum_{k=0}^{n} a_k so...

In an equation, the upper sum is Mi = 0+i(2/n) and the lower sum is mi = 0+(i-1)(2/n) So the question is why is it (i-1) for the lower sum and only i for the upper sum? Any help is highly appreciated! ^_^

Here are two cool functions defined by power series: \sum_{n=1}^{\infty}\frac{z^{n-1}}{(1-z^{n})(1-z^{n+1})}=\left\{\begin{array}{cc}\frac{1}{(1-z)^2},&\mbox{ if } |z|<1 \\\frac{1}{z(1-z)^2}, & \mbox{ if } |z|>1\end{array}\right. and...

If I want a series that sums to pi there are a lot of choices. I seem to recall that there is also at least one simple series that sums to a rational multiple of 1/pi, but I can't recall what it is. I managed to find a continued fraction expansion that gives 1/pi, but it didn't seem to...

I have been working on this problem for a while. I am supposed to prove that log 2 = \lim_{n \rightarrow \infty} \frac{1}{n+1} + \frac{1}{n+2} + ... + \frac{1}{2^n}. The problem is that I have a hard time figuring out how I am supposed to prove that something is equal to a transcendental...

I'm an EE student currently taking a Systems & Signals class. I've been searching high and low for information about convolution sums and convolution integrals. (Currently using the Haykin/Van Veen text). Here's my problem: I'm not grokking how the original input signal morphs into the output...

I was trying to help a friend do the following problem Prove with induction Sum of i^5 from 1 to n = \frac{n^2(n+1)^2(2n^2+2n-1}{12} we got it to \frac{(k+1)^2(k+2)^2(2k^2+4k+2)+2k+1}{12} but we can't seem to get it to go back to the orginal equation when you substitue k+1...

I need to know the names of theorems related to the following two problems: 1. What is the maximum sum less than 1 but more than 0 that can be formed from \frac{1}{p} + \frac{1}{q} + \frac{1}{r}, where p, q and r are positive integers? 2. What is the maximum perimeter and area of an...

Compute the following: \sum_{n=1}^{+\infty} \frac{1}{n^{2}} =...?? \sum_{n=1}^{+\infty} \frac{1}{n^{4}} =...?? .LINKS TO WEBPAGES WITH SOLUTIONS ARE NOT ALLOWED! :-p Daniel.

How many sums of money can be made from two pennies, four nickles, two quarters, and five dollar coins? The answer is 269 I've tried this question several ways, but cannot get the right answer. Tahnks for your help.

here is my problem: find the upper and lower sums for the region bounded by the graph of f(x) = x^2 and the x-axis between x=0 and x=2. I understand what this problem is asking but i don't understand how to compte the left and right endpoints. the left endpoint is the following...

If I have a problem like (N) sigma (K=0) Cos(Kpi) can I just move the sigma sign inside the brackets? like Cos(pi Sigma K) just wondering because I have this on an assignment problem and we didn't learn it in class and the textbook doesn't cover it either. If I can move it inside...

Please tell me if I am doing the summation of rectangular areas wrongly. Using summation of rectangles, find the area enclosed between the curve y = 3x^2 and the x-axis from x = 1 to x = 4. Now, before I answer the way it asks, I want to use antidifferentiation first to see what I should...

A little while ago I noticed a pattern in the sums of the digits of perfect squares that seems to suggest that: For a natural number N, the digits of N^2 add up to either 1, 4, 7, or 9. ex: 5^2 = 25, 2+5 = 7 In some cases, the summation must be iterated several times: ex: 7^2 = 49...

Consider this sequence: 1, 2, 3, 4, 5, ... We can calculate the n'th term of the sequence by the function t(n) = n. We could define s(n), the sum to n terms, recursively as s(1) = 1 and s(n) = n + s(n-1). The time bound of this procedure is O(n), but it isn't efficient because we can...

ive been searching for ages on how to do Reimann sums, and none of it makes any sense to me compared to other forms of integration. My problem is i have to use Riemann Sums to estimate the area under the curve f(x) = x^2 between x=2 and x=12. Any help would be hugely appreciated, Thanks

Which functions have the property that some upper some equals some (other) lower sum? Constant functions obviously do. Odd functions do in some cases. Even functions don't. Step functions won't (unless we restrict our consideration to an interval where it is constant). In fact it seems...

Hi all! I was wondering which method one should use to find the actual sum of an infinite series. I know how to find the sum of a geometric series (if it converges), but how could I find the sum for, for instance \sum_{n=0}^\infty\left(\frac{n+5}{5n+1}\right)^n I know that it converges...

Reimann sums, okay. How about a "Reimann product"? An integral is a sort of "continuous sum". Very roughly, the sum &Sigma;k f(xk) &Delta;x goes over to the integral &int;f(x)dx when the number of terms becomes infinite while &Delta;x goes to zero. What about a similar "continuous...

interesting inequality involving sums Which is the smallest n-natural number- for this inecuation: &#8721k=2n {1/[k * ln(k)]} &#8805 20 Any ideas?