Sums Definition and 349 Threads

Is there a general way to express, for instance, the coefficient of the order $x^j$ term in the expression $$\frac{\Sigma_{n}^{\infty}a_nx^n}{\Sigma_{m}^{\infty}b_mx^m}$$ ? Basically I am working with a quotient of two infinite power series and I want to know the term in this quotient that is...

Homework Statement I am trying to wrap my head around what it means to find an explicit formula for the sequence of partial sums. Question: Find an explicit formula for the sequence of partial sums and determine if the series converges. a) sum from n=1 to n=infinity of 1/(n(n+1)) Homework...

so, continuous signals as sums of weighted delta functions can be represented like this: if you switch order of some variables you get ∫x(τ)δ(-τ+t)dτ, and since,I presume, Dirac delta "function" is even I can write it like this ∫x(τ)δ(-(-τ+t))dτ=∫x(τ)δ(τ-t)dτ=x(t) and we got ourselves a...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 6.1 The Jacobson Radical ... ... I need help with the proof of Proposition 6.1.4 ... Proposition 6.1.4 and its proof read as follows: In the above proof from Bland we read:"... ... If $$i_1 \ : \ M_1...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 6.1 The Jacobson Radical ... ... I need help with the proof of Proposition 6.1.4 ...Proposition 6.1.4 and its proof read as follows: In the above proof from Bland we read:"... ... If ##i_1 \ : \ M_1...

Homework Statement Homework Equations N/A The Attempt at a Solution I proved the first part of the question (first quote) and got stuck in the second (second quote). I defined Im(E1) as U and Im(E2) as W and proved that v=u+w where v ∈ V, u ∈ U and w ∈ W. After that however I got stuck at...

Hi is there a way or algorithm to find the sigma notation of sums in which the sums do not have an apparent general form?

Homework Statement Find the sum of the following series: $$ \left( \frac 1 2 + \frac 1 4 \right) + \left( \frac 1 {2^2} + \frac 1 {4^2} \right) +~...~+ \left( \frac 1 {2^k} + \frac 1 {4^k} \right) +~...$$ Homework Equations $$ \sum_{n = 1}^{\infty} \left( u_k+v_k \right) = \sum_{n =...

Svein submitted a new PF Insights post Using the Fourier Series To Find Some Interesting Sums Continue reading the Original PF Insights Post.

Homework Statement [/B]Homework Equations an= bn - bn+1 which is already in the problem The Attempt at a Solution [/B] i did partial fractions but then i got stuck at 16/12 [4n-5] - 16/12 [4n+7] that part about bn confuses me please someone explain in detail

Homework Statement Determine ##\int_{0}^{2}\sqrt{x}dx## using left riemann sums Homework Equations ##\int_{a}^{b}f(x)dx = \lim_{n\rightarrow \infty}\sum_{i=0}^{n-1}(\frac{b-a}{n})f(x_i)## The Attempt at a Solution [/B] ##\frac{b-a}{n}=\frac{2-0}{n}=\frac{2}{n}## ##\int_{0}^{2}\sqrt{x}dx =...

Consider the Taylor series expansion of ##e^{-x}## as follows: ##\displaystyle{e^{-x}=1-x+\frac{x^{2}}{2}-\frac{x^{3}}{6}+\dots}## For ##x>0##, the partial sums ##1##, ##1-x##, ##\displaystyle{1-x+\frac{x^{2}}{2}}## bound ##e^{-x}## from above and from below alternately. How do I prove this?

hi, I'm solving solving a problem about sums of zeta function and I'm come to the following conclusion $$\sum _{n=2}^{\infty }{\frac {\zeta \left( n \right) }{{k}^{n}}}= \sum _{s=1}^{\infty } \left( {\it ks} \left( {\it ks}-1 \right) \right) ^{-1}=\int_{0}^{1}\!{\frac {{u}^{k-2}}{\sum...

Please take a look of the photo. In the middle part, it says For each I, by division and gets the following results. Please further explain to me how to get the result by division. The photo is attached. Attempt: 1. Using f(x)=α0(x-α1)(x-α2)...(x-αn) form. If it is divided by x-αi , there...

https://wikimedia.org/api/rest_v1/media/math/render/svg/a7fd3adddbdfb95797d11ef6167ecda4efe3e0b9 https://en.wikipedia.org/wiki/Lorentz_force#Lorentz_force_in_terms_of_potentials How to write this formula in terms of sums and vector components? What is ##v\cdot\nabla## ? I think it is some...

I have column from E5 -->E35 which contains values for each box. another column from F5-->F35 contains values for each box I have a sums column, H5-->H35,, which contains values for the sums between the E and F boxes (E5+F5=H5 ;;; E6+F6=H6 etc...) I want a division column inside J5-->J35 which...

Hi Community, I have the following question: I have done basic solving of limits and also of Riemann sums but never had to do them in the same question. Would I be correct in saying that I need to solve for the Riemann sum first then take the limit of the integral? Cheers Nemo

Homework Statement Using: \mathcal{L}\big\{t^n\big\}=\frac{n!}{s^{n+1}}\text{for all s>0} Give a formula for the Laplace transform of an arbitrary nth degree polynomial p(t)=a_0+a_1t^1+a_2t^2+...+a_nt^n Homework Equations \mathcal{L}=\lim_{b\rightarrow\infty}\int_{0}^{b}p(t)e^{-st}dt The...

Hi and sorry for bad english.I want to know if these properties are truehttps://gyazo.com/88c471f4bb9989b67a390c372f2c72fe and https://gyazo.com/85f4110664db6831576012debaf3a778 I did not find these properties in any place. so I guess it will be obvious or are incorrect, if incorrect I would...

My Calculus 2 teacher's lecture slides say: Many of the functions that arise in mathematical physics and chemistry, such as Bessel functions, are defined as sums of series. I was just wondering how this was different from the basic functions that we've already worked with. Are they not...

I've been working on a problem for a couple of days now and I wanted to see if anyone here had an idea whether this was already proven or where I could find some guidance. I feel this problem is connected to the multinomial theorem but the multinomial theorem is not really what I need . Perhaps...

Homework Statement Suppose that AB = AC for matrices A, B, and C. Is it true that B must equal C? Prove the result or find a counterexample. Homework Equations Properties of matrix multiplication The Attempt at a Solution AC = A(D + B) = AD + AB = 0 + AB = AB ? Can someone help me...

Homework Statement Homework EquationsThe Attempt at a Solution So I am tasked with answer #3 and #4. I have supplied the indicated parenthesis of 8 also with the image. Here is my thinking: Take the Fourier series for |sin(θ)|. Let θ = 0 and we see a perfect relationship. sin(0) = 0 and...

In my book, applied analysis by john hunter it gives me a strange way of stating an infinite sum that I'm still trying to understand because in my calculus books it was never described this way. It says: We can use the definition of the convergence of a sequence to define the sum of an...

Homework Statement ∑n!/(3*4*5...*n) s1=1/3 sn=1/3+2/(4*3)+3!/(5*4*3)+...+n!/(3*4*5*...n) so i multiplied the sum with 1/2sn=1/6+1/(4*3)+1/(5*4)+1/(6*5)...+1/((n+2)(n-1)) got blocked here,i don't know how to continue, help please

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 need help with Exercise 1.2.12 ... Exercise 1.2.12 reads as follows: https://www.physicsforums.com/attachments/5102 Can someone please help me to get started on this problem ...

hello, sorry for bad English, i have a question. if we consider the following equations and we take natural values note that tend 2 x-1=0 -----------------> x = 1 x^2-x-1=0 ----------------->...

Homework Statement Find the upper, lower and midpoint sums for $$\displaystyle\int_{-3}^{3} (12-x^{2})dx$$ $$\rho = \Big\{-3,-1,3\Big\}$$ The Attempt at a Solution For the upper: (12-(-1)^2)(-1-(-3)) + (12-(-1))(3-(-1)) =74 For the lower: (12-(-3)^2)(-1-(-3))+(12-3)(3-(-1)) =42 For midpoint...

Just want to see if I actually understand what these all mean. Partition: is like the x-coordinate values, also gives the number of times the graph was chopped up. We need them in order to find the distance or length of each rectangle. The distance is found by taking the further point minus...

Homework Statement Problem Consider a random experiment with a sample space S={1,2,3,⋯}. Suppose that we know: P(k) = P({k}) = c/(3^k) , for k=1,2,⋯, where c is a constant number. Find c. Find P({2,4,6}). Find P({3,4,5,⋯}) I am primarily interested in part 1, finding C. The rest...

Homework Statement Hi, I am reviewing a practice exam for my course and I am a bit stuck. "Assume that a sequence of partial sums (s_n) converges, can we also then say the sequence a_n is convergent? Does this statement go both ways? Answer: Yes, yes" The Attempt at a Solution On our exam...

∫(4−7x)dx ====> Integral is from 1 to 8 Similar question but for some reason I can't get the answer after following all the steps

sorry new to this site. Can someone please help me with this? I have tried for such a long time and have yielded no correct answers. ∫(3−5x)dx ======> integral is from 1 to 7 We have n rectangles, so what I did first was found the change in x, which was 6/n which is the width of the...

Hi brand new to the site. I keep on having a syntax error when I run the code below on my casio fx-cg10. Btw I also put a display triangle on the last m as well

Possibly a difficult question, but I've never found a discussion on the topic. Thanks

I tackle the following game analysis: 2 players, two 6-sided dice. Bigger sum of points win. First roller has an advantage, as he wins even if 2nd player's dice sum equals to his. As the game is played with doubling cube (potentially increasing the odds before any roll), I tried to enumerate...

I am reading Manfred Stoll's book: Introduction to Real Analysis. I need help with Exercise 2(a) from Stoll's Exercises 6.2 on page 229 ... Exercise 2 reads as follows: I was somewhat puzzled about how to do this exercise ... BUT ... even more puzzled when I read Stoll's hint for solving the...

Question: Convert to Sum of Products (p+q’+r+s’). (p’+q+r+s’). (p’+q’+r+s’) = ((p+q’+r+s’)’ + (p’+q+r+s’)’ + (p’+q’+r+s’)’)’ (Demorgans each Sum-Term) = ((p’.q.r’.s) + (p.q’.r’.s) + (p.q.r’.s))’ (Factor p.r’.s) = ((p’.q.r’.s) + (p.r’.s).(q’ + q))’ (Inverse Law) = ((p’.q.r’.s) + (p.r’.s))’...

Homework Statement Find the sum of the following series: Σ n*(1/2)^n (from n = 1 to n = inf). Homework Equations I know that Σ r^n (from n = 0 to n = inf) = 1 / (1 - r) if |r| < 1. The Attempt at a Solution [/B] I began by rescaling the sum, i.e. Σ (n+1)*(1/2)^(n+1) (from n = 0 to n =...

Homework Statement The diagram shows 3 vector all of equal length. Which statement is true? a. A+B=A-C b. A+B=B-C c. A-B=2A-C d. A-B=2A+C e. 2A+2B=2C Homework Equations None The Attempt at a Solution I just added them in my head, and thought that e. 2A+2B=2C would also work. Why doesn't it?

I am reading D.G. Northcott's book: Lessons on Rings and Modules and Multiplicities. I am currently studying Chapter 2: Prime Ideals and Primary Submodules. I need help with a result that Northcott quotes and proves on page 80 regarding sums and products of ideals. The relevant text from...

Homework Statement I have that, for n ∈ ℕ, (-6/π) ∑n even (cos(nx))/(n2-9) is equivalent to (-6/π) ∑∞n=1 (cos(2nx))/(4n2-9). I don't understand how the two sums are equivalent to each other. Homework Equations I honestly have no idea what may be relevant, other than what is above. The...

Hi! (Wave) I want to write an algorithm, that counts the sum of the keys of the nodes of a binary tree, without the use of globals and statics. That's what I have tried: S(NODE *P){ if (P==NULL) return 0; int m=P->data+S(P->left); int n=m+S(P->right); return n; Could you tell me if it...

1). Find the fourth term of the sequence of partial sums for the given sequence. {5+ 3\2 n} 2). A bicycle rider coasts downhill, traveling 7 feet the first second. In each succeeding second, the rider travels 6 feet farther than in the preceding second. If the rider reaches the bottom of the...

Homework Statement You are given the following two sums: 1+2+3+...+50 = 1275 and 1+2+3+...+100 = 5050 Using these results, find the following sums. (no long methods allowed) a) 51+52+53+...+100 b) 2+4+6+...+100 c) 1+3+5+...+99 d) 1-2+3-4+5-...+99-100 e) 100.01-100.02+100.03-100.04+... -101...

Hi guys, Suppose I have the function x = a + b -1 where a, b have expected values of 0.5 each. What is the expected value of x? is it 0.5 + 0.5 -1 = 0? or is it just 0.5 + 0.5? Secondly, suppose the same equation as above, x = a + b -1. If the variance of both a and b is 1/12, what is the...

In Paul E. Bland's book: Rings and Their Modules, the author defines the external direct sum of a family of $$R$$-modules as follows: Two pages later, Bland defines the internal direct sum of a family of submodules of an $$R$$-module as follows: I note that in the definition of the external...

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...