Sums Definition and 370 Threads

Task in real analysis: P is a uniform partition on [0, π] and is divided into 6 equal subintervals. Show that the lower and upper riemann sums of sin (x) over P is lesser than 1.5 and larger than 2.4 respectively. My attempt at the solution: The greates value and the least value of sin x over...

Does anyone know how to express a given function as a sum in mathematica using sigma notation? For example, I know how to make ##e^x = 1 + x^2/2 + x^3/6 + x^4/24...## but how would I have mathematica write it as ##e^x = \Sigma_{n=0}^{\infty} x^n/n!##? Thanks a ton!

I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K). In Chapter2: Direct Sums and Short Exact Sequences we find Exercise 2.1.6 part (iii). I need some help to get started on this exercise. Exercise 2.1.6 reads as follows: I am...

I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K). In Chapter2: Direct Sums and Short Exact Sequences in Sections 2.1.14 and 2.1.15 B&K deal with ordered index sets in the context of direct sums and products of modules. Although it...

I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K). I an trying to gain a full understanding of direct products and external direct sums of modules and need some help in this matter ... ... B&K define the external sum of an arbitrary...

I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K). In Chapter2: Direct Sums and Short Exact Sequences in Sections 2.1.11 and 2.1.12 B&K deal with infinite direct products and infinite direct sums (external and internal). In Section...

Hello guys, since I am new at sums and multivariable calculus I faced a problem when I stumbled upon this: \sum_{r=0}^{k} \binom{n}{4r+1} x^{n-4r-1} y^{4r+1} = \sum_{r=0}^{b} \binom{n}{4r+3} x^{n-4r-3} y^{4r+3} Well, the problem is that I don't know if it's possible to put a limit in every...

Consider a one dimensional harmonic oscillator. We have: $$\hat{n} = \hat{a}^{\dagger} \hat{a} = \frac{m \omega}{2 \hbar} \hat{x}^2 + \frac{1}{2 \hbar m \omega} \hat{p}^2 - \frac{1}{2}$$ And: $$\hat{H} = \hbar \omega (\hat{n} + \frac{1}{2})$$ Let's say we want to measure the total...

Exercise 2.1.6 (i) of Berrick and Keating's book An Introduction to Rings and Modules reads as follows:Let M = M_1 \oplus M_2, an internal sum of right R-modules, and let \{ \sigma_1 , \sigma_2, \pi_1 , \pi_2 \} be the corresponding set of inclusions and projections. Given an endomorphism \mu...

Can someone explain to me why in this equation (attached) where ρ(t)=\sumδ(t-ti) , dirac funtion. in the left side we have the sum over h(t-ti) instead of the sum over h(ti) ? It seems to me that the integral would work summing 1*h(t1)+1*h(t2)+...+1*h(ti) for all ti smaller than t.

I am reading Chapter 2: Vector Spaces over \mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C} of Anthony W. Knapp's book, Basic Algebra. I need some help with some issues regarding infinite direct sums and products and indexed families of sets ... ... On page 62, Knapp introduces direct...

I am reading Chapter 2: Vector Spaces over \mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C} of Anthony W. Knapp's book, Basic Algebra. I need some help with some issues regarding Theorem 2.30 (regarding an isomorphism between external and internal direct sums) on pages 59-60. Theorem 2.27 and...

Hey guys, I'd appreciate some help for this problem set I'm working on currently The u-substitution for the first one is somewhat tricky. I ended up getting 1/40(u)^5/2 - 2 (u) ^3/2 +C, which I'm not too sure about. I took u from radical 3+2x^4. For the second question, I split the integral...

I've been thinking about the Central Limit Theorem and by my understanding it states that the sum of randomly distributed variables follows approximately a normal distribution. My question is if you have, say, 100 uniformly distributed variables that range from 0 to 10, their sum has to be...

How could I find the lim as n-> infinity of the expression I attached? The only way I could find was to express it in terms of a definite integral. Integral of xe^(-2x) from 0 to 1. What is the other way?

digits sum $A\in ${1,2,3,4,----2013} a number p is picking randomly from A if $q_1$=the digits sum of p if $q_2$=the digits sum of $q_1$ ------ continue this procedure until the digits sum=1 then stop How many numbers we can pick from A ,and meet the requirement ?

I can't grasp the underlying process on how this is working. n/2(f+l) = algorithm sum of all integers n= number of all integers f= first integer l= last integer Example: 1, 2, 3, 4 4/2(1+4) 2(5) = 10 I know how to do it, but I don't really understand how to actually do it. Am I...

Is there a "Leibnitz theorem" for sums with variable limits? Wikipedia says that if we want to differentiate integrals where the variable is in the limit and in the integrand, we can use Leibnitz theorem: But what if I need to integrate a function defined like this:\Sigma_{I(x)}[f(x,t)]...

I was reading the Wikipedia article about the sum 1+2+3+4+..., and I saw this explanation: c = 1+2+3+4+5+6+... 4c = _4__+8__+12+... -3c = 1-2+3-4+5-6+... link: http://en.wikipedia.org/wiki/1_+_2_+_3_+_4_+_%E2%8B%AF My question, as one who hasn't worked with infinite sums: Why are you...

Hello. I'm not sure what type of problem this is that I'm trying to solve. Any pointers would be greatly appreciated. Suppose you have a list of numbers and you want to form them into, say, 4 groups such that the sum of each group is, as nearly as possible, equal to the sums of each of the...

Ib= (-12.4-j6.88)-(9.57-j6.38) Ib= -12.4-j6.88-9.57+j6.38 Ib=(-22-j0.5)A Why does the -j6.38 become +j6.38? Ia=(9.57-j6.38)-(0.454+j13) Ia= 9.57-j6.38-0.454-j13 Why does the +J13 become -J13? Thanks

Homework Statement "In a geometric sequence, the sum of t7 and t8 is 5832, the sum of t2 and t3 is 24. Find the common ratio and first term." Homework Equations d = t8/t7 or t3/t2 tn = a * rn-1 The Attempt at a Solution So I thought of developing a system of equations then solving...

Prove that \sum_{k=0}^{n} \sin\left( \frac{k \pi}{n} \right) = \cot \left( \frac{\pi}{2n} \right)

In R. Y. Sharp "Steps in Commutative Algebra", Section 2.23 on sums of ideals reads as follows: ------------------------------------------------------------------------------ 2.23 SUMS OF IDEALS. Let ( {I_{\lambda})}_{\lambda \in \Lambda} be a family of ideals of the commutative ring R . We...

We all know about the sum of the infinite series, 1 + 1/1! + 1/2! + 1/3! + ... to 1/inf! = e What other series do we have that are equal to 'e'?

Homework Statement My apologies in advance for the messiness of the equations; the computers available to us do not correctly process the LaTex code. I am tasked with estimating the area under the curve f(x)=x2+1 on the interval [0,2] using 16 partitions. Online calculators and my...

Show that $$ \begin{align*} \sum_{n=0}^\infty \binom{2n}{n}^3 \frac{(-1)^n}{4^{3n}} &= \frac{\Gamma\left(\frac{1}{8}\right)^2\Gamma\left(\frac{3}{8}\right)^2}{2^{7/2}\pi^3} \tag{1}\\ \sum_{n=0}^\infty \binom{2n}{n}^3 \frac{1}{4^{3n}}&= \frac{\pi}{\Gamma \left(\frac{3}{4}\right)^4}\tag{2}...

Here are the questions: I have posted a link there to this thread so the OP can see my work.

I am really having trouble understanding how upper and lower sums are calculated. In the equations Ʃf(Mi)Δx and Ʃf(mi)Δx, what do Mi and mi represent?

Hi, Homework Statement I am asked to evaluate the following sum S=Sigma(n=0 to N) x^n/n! (namely, e^x as n->Inf) for N=10:10:100 and x=10, so that every element S(i) is a partial sum which approximates function e^x with different accuracy. Below is my code, which doesn't work. Homework...

Need someone to check my answer please. Consider a 4 input, 1 output digital system (W,X,Y,Z, and f respectively) . Design a POS circuit with any number of inputs such that f(W,X,Y,Z) = M(0,2,4,9,13) + D(6,14). First fill in the Truth Table, then find the minimum product of sums equation using...

Write out the minimal Product of Sums (POS) equation with the following Karnaugh Map. Just need someone to check my work please. I am questioning my self on my grouping. Did I group correctly or should I have grouped the bottom left 0 and D versus the 0 in the group of 8? Thanks for your time...

Awesome thanks.. Mind checking this as well? Minimize Sum of Products equation given the following K map. My Answer: \bar{y} \bar{w} + wx + y\bar{z}w + yw\bar{x}

Just need someone to check my work. Couldn't find the problem via Google. $f$(W,X,Y,Z) M (0,1,2,7,12,15) + d(3,13). 1)Find the minimum Product of Sums equation using a K-Map. 2)Draw a schematic of a minimized circuit implementing the logic using NOR gates. 1) My Answer: (\bar{w} +...

The identity map on the direct sum of V1 and V2 would be i1 composed with p1 + i2 composed with p2. Would such an identity map exist for an infinite direct sum? And an analogous mapping for a direct product?

I am reading John Dauns book "Modules and Rings". I am having problems understanding the notation of Section 1-2 Direct Products and Sums (pages 5-6) - see attachment). In section 1-2.1 Dauns writes: ================================================== ====== "1-2.1 For any arbitrary...

I am reading John Dauns book "Modules and Rings". I am having problems understanding the notation in section 1-2 (see attachment) My issue is understanding the notation on Section 1-2, subsection 1-2.1 (see attachment). Dauns is dealing with the product \Pi \{ M_i | i \in I \} \equiv \Pi M_i...

Explain why the sum, the difference, and the product of the rational numbers are rational numbers. Is the product of the irrational numbers necessarily irrational? What about the sum? Combining Rational Numbers with Irrational Numbers In general, what can you say about the sum of a rational...

Prove that: \prod_{j=2}^n\left(1-\frac{1}{\sum\limits_{k=1}^j(k)} \right)=\frac{n+2}{3n}

So I was trying to see if \Sigmaln(\frac{n}{n+1}) diverges or converges. To see this I started writing out [ln(1) - ln(2)] + [ln(2) - ln(3)] + [ln(4) - ln(5)] ... I noticed that after ln(1) everything must cancel out so I reasoned that the series must converge on ln(1) which equals ZERO...

is it also possible to transform any these kinds summation to any product notation: 1. infinite - convergent 2. infinite - divergent 3. finite (but preserves the "description" of the sequence) For example, I could describe the number 6, from the summation of i from i=0 until 3. Could I...

Homework Statement Differentiate f(x) = x^{1/2} - x^{1/3} Homework Equations f(x) = f'(x)- g'(x) The Attempt at a Solution I am a little stuck about what to do after the first couple steps. Here is my attempt. f(x) = x^{1/2} - x^{1/3} f'(x) = (x^{1/2})' -(x^{1/3})' =...

I'm having trouble understanding this. Suppose I have a sum ##\displaystyle \sum_{i=1}^{n}\left[\int_{a}^{b} f(t) dt\right]##, where f(t) depends on both n and i. Under what conditions could this expression be equal to the same expression with the integral and the summation in reversed order...

Homework Statement Let \{a_{n,k}:n,k\in\mathbb{N}\}\subseteq[0,\infty). Prove that \sum\limits_{n=0}^{\infty}\sum\limits_{k=0}^{n}a_{n,k}=\sum\limits_{k=0}^{\infty}\sum\limits_{n=k}^{\infty}a_{n,k}. Homework Equations The Attempt at a Solution I am pretty certain that the claim is...

I have a set of questions concerning the perennial sum \large \sum_{k=1}^{n}k^p and its properties. 1. For p \ge 0, the closed form of this is known (via Faulhaber's formula). I know little about divergent series, but I've read that in some sense there exists a value associated with these sums...

We are already introduced to finding the value of definite integral by the anti-derivative approach \int_{a}^{b}f(x) dx = F(b) - F(a) In this approach we find the anti-derivative F(x) of f(x) and then subtract F(a) from F(b) to get the value of the definite integral Reimann Sums...

Evaluate the following discrete-time convolution: y[n] = cos(\frac{1}{2}\pin)*2^{n}u[-n+2] Here is my sloppy attempt: y[n] = \sumcos(\frac{1}{2}\pik)2^{n-k}u[-n-k+2] from k = -∞ to ∞ = \sumcos(\frac{1}{2}\pik)2^{n-k} from k = -∞ to 2 We can re-write the cos as...

Find the limit limn→∞∑i=1 i/n^2+i^2 by expressing it as a definite integral of an appropriate function via Riemann sums ...?

Hi all, I was just looking at the U.S. electoral map, and I was wondering if there could possibly be a tie in presidential elections (the answer is probably no). I tried to think of an efficient algorithm to answer this question, but due to my limited intelligence and imagination, all I...

when using the reimann integral over infinite sums, when is it justifiable to interchange the integral and the sum? \int\displaystyle\sum_{i=1}^{\infty} f_i(x)dx=\displaystyle\sum_{i=1}^{\infty} \int f_i(x)dx thanks ahead for the help!