Summation Definition and 610 Threads

Hi, let's take the sum: $\displaystyle \sum_{n=1}^{\infty}\frac{1}{9n^2 + 3n - 2}$ $\implies 9n^2 + 3n - 2 = 9n^2 + 6n - 3n - 2 = 3n(3n + 2) - (3n + 2) = (3n - 1)(3n - 2)$ The simplest way would be to use partial fractions, and then convert this into a telescoping series. Which makes the sum...

Homework Statement I'm currently trying to follow a derivation done by Shankar in his "Basic Training in Mathematics" textbook. The derivation is on pages 343-344 and it is based on the solution to the two dimensional heat equation in polar coordinates, and I'm not sure how he gets from one...

Hi, I'm just trying to evaluate a series and would just appreciate if someone could either verify or correct me work. Essentially, I have a series that I've produced: -[(t^2)/2 + (t^5)/(2x5) + (t^8)/(2x5x8) + ...] = - *sum from n = 0 to infinity* [(t^(3n+2))/(3n+2)!] = -e^t Sorry for the...

Hi I just wanted to check my approach. I have spectrum I have peak at 10Hz another at 20Hz and a third at 30Hz. The amplitudes are 1000, 500, 250. I want to recreate the signal by summing sine waves. I assume that I will therefore take A1 = 1; A2 = 0.5; A3 = 0.25; I will then let y =...

Hi Can I ask a question please. I have an equation that involves the summation over some indices, for example. A^αβ B_αγ = C^β_γ Say that I don't know Β_αγ , and want to make this the subject of the equation, how is this done? Thanks Peter

Given $a_1,...,a_n$, find the minimum value of $\sum_{i}^{n}(x-a_i)^2$ No idea how to do it. I was thinking maybe when $x-a_i=0$, but I think $x$ is constant so it won't work...unless the series $a_n$ is constant too...Tiny hint please :D?

Homework Statement Show that the expected number of successes in n Bernoulli trials w probability p of success is <x> = np Homework Equations The Attempt at a Solution So I get the right answer which is this: E\left( x\right) =\sum _{x=0}^{n}x\left( \begin{matrix} n\\...

Prove that $\displaystyle\left(\sum_{k=1}^{n} \sqrt{\dfrac{k-\sqrt{k^2-1}}{\sqrt{k(k+1)}}}\right)^2\le n\sqrt{\dfrac{n}{n+1}}$, where $n$ is a positive integer.

In chapter 2.2 of Feynman's book on QFT, he states that the probability amplitude of a particle going from a to b is the sum of contributions from all paths, and that each path contributes the same amplitude, but with a different phase. My question is, why does Feynman state that this is the...

Hey, I've begun going through a book called "An introduction to geophysical exploration" by Phillip Kearey and Michael Brooks and I've come across a problem I can't for the life of me see how they got their answer. Essentially, given an input function gi (i = 1,2... m), and a convolution...

$S_k:5\cdot 6 +5\cdot 6^2+5\cdot 6^3+ ...+5\cdot 6^k=6(6^k-1)$$S_k:5\cdot 6 +5\cdot 6^2+5\cdot 6^3+ ...+5\cdot 6^k+ 5\cdot 6^{k+1}=6(6^k-1)+5\cdot 6^{k+1}$ what do i do now? to prove $S_{k+1}$

Homework Statement I hope this is the right forum for this question. I am starting to self-teach calculus, could you help me shape my problem? I am trying to use wolfram: I know that if I integrate an equation say: 5/\sqrt x, I will get the area underneath that curve...

Hello everyone, I've been working on an area summation problem in my book for quite a bit and I can't solve it. Find the area under the straight line y=2x between x = 1 and x = 5 The book shows the answer as 24 and Maple does as well, but I'm not getting 24, I'm getting 8. Area summation formula...

Prove the following: $$\sum_{k=1}^n \left(2^k\sin^2\frac{x}{2^k}\right)^2=\left(2^n\sin\frac{x}{2^n}\right)^2-\sin^2x$$

Evaluate the following: $$\Large \sum_{k=1}^{\infty} (-1)^{\left\lfloor \frac{k+3}{2} \right\rfloor} \frac{1}{k}$$

Hello, Some of you may know this equation and I need help solving it. Delta ti is the time in days at a certain temperature (0 - 80). T(Delta ti) is the temperature during Delta ti. The answer is supposed to be an age in days but my tries have given me answers that are below 1, which doesn't...

r is rational ,and all the roots of equation: $rx^2+(r+2)x+r-1=0$ are integers please find :$\sum r^3$

Homework Statement I have a set of data (i, yi). A polynomial fit of 1st degree would be y=ai+b, right? If I have c=Σ(i2*yi) is it correct to substitute y=ai+b inside the summation? Homework Equations The Attempt at a Solution

Homework Statement I have an equation in the general form: const=b*∑i2yi+∑f(I)f(y)...) where const,b are known constants.I have a general question.Is it possible from equations like this to identify how the ys should be distributes so as the const takes a specific value, e.g const=0.05? What...

The following identities are true? $$\frac{d}{dx} \sum_{u_0}^{u_1}f(x,u)\Delta u = \sum_{u_0}^{u_1}\frac{d}{dx}f(x,u)\Delta u$$ $$\int \sum_{u_0}^{u_1}f(x,u)\Delta u dx = \sum_{u_0}^{u_1}\int f(x,u)dx\Delta u$$

I need to solve the equation for x, where a is a known constant and . The bs are known too. What i need to do is sto find for which xs I'll have a specific value of a, eg a=0.5, i.e. solve for x and substitute the a. I believe that the result will be a group of xs and not a single...

Could someone please explain why the following sum simplifies to the following? = As far as I can see, this sum does not correlate to the formula for incomplete gamma function as a sum. I'd appreciate any help as the incomplete gamma function is somewhat beyond the scope of my current...

Say for some general function f(x), and g(x) = ∑x=0∞ f(x) (assuming function is defined) Is there a way to find the zeroes of g(x)? Is there any relationship between the zeroes of f(x) and g(x)? Sorry if this question is poorly asked, i just began learning about summations and infinite series...

If exist a connection between the infinitesimal derivative and the discrete derivative $$d = \log(\Delta + 1)$$ $$\Delta = \exp(d) - 1$$ exist too a coneection between summation ##\Sigma## and integration ##\int## ?

Hello, i'm doing some practice problems using Gauss' law, but I feel like my work is 'sloppy'. I'll show an example, where I think I get the right answer, but it feels like I'm neglecting to treat the summation properly, or perhaps I don;t quite understand why what I'm doing is fine...

Homework Statement Find the magnitude and direction of a resultant force equivalent to the given force system and locate its point of application on the slab. The Attempt at a Solution So I summed the forces to get -1400 N, or a 1400 N force downward (the book agrees with that). Why is the...

Homework Statement find the general formula to calculate the sum Homework Equations 1+11+111+1111+11111+....upto n terms The Attempt at a Solution 100 + (101+100) + (102+101 + 100) + (103 + 102+101 + 100) + ... ==> (100+100+100+...upto n terms) + (101+101+101+...upto n-1 terms)...

From an exercise set on the summation convention: X and Y are given as [Xi] = \begin{pmatrix} 1\\ 0\\ 0\\ 1\end{pmatrix} and [Yi] = \begin{pmatrix} 0\\ 1\\ 1\\ 1\end{pmatrix} There are a few questions involving these vectors. The one I am stuck on asks to compute XiYj . It may be necessary...

Problem: Evaluate: $$\left[\sum_{n=1}^{\infty} \sum_{k=2}^{2014} \frac{1}{n^k}\right]$$ where $[x]$ denotes the floor function.Attempt: I can see that the above can be written as: $$\sum_{n=1}^{\infty} \frac{1}{n^2}+\frac{1}{n^3}+\frac{1}{n^4}+\cdots + \frac{1}{n^{2014}}$$ $$=\sum_{n=1}^{\infty}...

Simplification -- complicated summation involving delta functions Homework Statement \frac{1}{\sqrt{(2^3)}}\sum[δ(k+1)+δ(k-1)]|k> for k=0 to 7 Homework Equations The Attempt at a Solution I am trying to simplify the above expression. I get \frac{1}{∏*\sqrt{(2^3)}} |1>, which is...

Consider this Summation: ∑cos^2 (∏*n / 4) limits: -N to N when I type that on wolframAlpha I get the following: http://www.wolframalpha.com/input/?i=summation+%281%2B+cos%28pi+n+%2F+2%29%29+from+-N+to+N I have no Idea how it was performed though. how Can I transform this...

Let $$\Large S_n=\sum_{k=1}^{4n} (-1)^{\frac{k(k+1)}{2}}k^2$$ Then $S_n$ can take the value(s) A)1056 B)1088 C)1120 D)1332

i'm kinda confused regarding summation so I'm hoping someone can help me figure this out and explain to me why it is the way it is trace(AB*) = ? in summation form * = adjoint = conjugate and transpose = transpose and conjugate assume both matrices are square mx of same size n x n...

Here is the question: I have posted a link there to this thread so the OP can view my work.

I came across some summation but have no idea how to simplify it. $\sum_{x=0}^{\infty} \binom{x+r-2}{r-2}(1-p)^{x}=p^{1-r}$ Why is it so?

Im trying to find a general formula I can store in my calculator that can find the number of onto (surjective) functions exists for a relation of when M is mapped to N. I can't seem to find a nice formula for it, but based on the below material I will show you what I have developed. From...

The title pretty much says it all, does anyone know of an infinite series summation that is equal to $$\sqrt{-1}$$?

Homework Statement I need a summation where the answer is 1 2 2 2 2 2 2 2 Homework Equations a(0) + sum(2*a(1) + 2*a(2) +2*a(3)) The Attempt at a Solution I unfortunately have no idea where to start, basically it is taking a symmetrical function from 0 to N-1. where the function...

Problem: Consider a function $f(n)$ defined as: $$f(n)=\sum_{r=1}^n (-1)^{r+1} \binom{n}{r} \left(\sum_{k=1}^r \frac{1}{k}\right)$$ Find the value of $$\sum_{i=1}^{\infty} (-1)^{i+1}f(i)$$ Attempt: I write $\sum_{k=1}^r (1/k)=H_r$. The sum I have to evaluate is $$f(1)-f(2)+f(3)-f(4)+\cdots$$...

Like in the integration, exist a formula to compute the summation by parts, that is: \frac{\Delta }{\Delta x}(f(x)g(x))=\frac{\Delta f}{\Delta x}g+f\frac{\Delta g}{\Delta x}+\frac{\Delta f}{\Delta x}\frac{\Delta g}{\Delta x}\sum \frac{\Delta }{\Delta x}(f(x)g(x))\Delta x = \sum \frac{\Delta...

$ n\in N$ $n<\sqrt n + \sqrt[3]{n} + \sqrt[4]{n}$ find :$ \sum n $

Problem: Evaluate $$\lim_{n\rightarrow \infty} \left(\sum_{r=1}^n (\arctan(2r^2))-\frac{n\pi}{2}\right)$$ Attempt: I tried evaluating the summation but couldn't. Had the problem involved $\arctan(1/(2r^2))$, I could rewrite it as $$\arctan\left(\frac{2r+1-(2r-1)}{1+(2r+1)(2r-1)}\right)$$ and...

I'm teaching a course using D. V. Schroeder, An Introduction to Thermal Physics, and there is a "derivation" in the book that is making me cringe a bit. I would like the opinion of mathematicians on the subject. Take a (continuous) degree of freedom ##q## from which you can get the energy...

Problem: Find the value of $$\lim_{n\rightarrow \infty} \sum_{r=0}^n \left(\frac{1}{4r+1}-\frac{1}{4r+3}\right)$$ Attempt: I tried writing down a few terms to see if the terms cancel but no luck there. I couldn't find any closed form for the summation. :( Next, I thought of converting it into...

hey pf! can someone explain to me what to do if presented with an equation like this: \sum_{i=1}^{n}A_i=i is this identical to stating A_i=i? either way, can you please explain. thanks! josh

Find the sum of the following series upto infinite terms: $$\cot^{-1}\left(\frac{5}{\sqrt{3}}\right)+\cot^{-1}\left(\frac{9}{\sqrt{3}}\right)+\cot^{-1}\left(\frac{15}{\sqrt{3}}\right)+\cot^{-1}\left(\frac{23}{\sqrt{3}}\right)+\cdots$$

Homework Statement Someone please check my work... :D If ##f(x)=\sqrt{x}+\sqrt{x+1}## , find the value of ##\frac{1}{f(1)}+\frac{1}{f(2)}+\frac{1}{f(3)}+...+\frac{1}{f(24)}## Homework Equations Summation of series, rationalizing the denominator. The Attempt at a Solution...

So the title pretty much says it all, what other infinite series summations do we have for Pi^2/6 besides, $$\sum_{n=1}^{\infty} 1/n^2$$ ***EDIT*** I should also include, $$\sum_{n=1}^{\infty} 2(-1)^(n+1)/n^2$$ $$\sum_{n=1}^{\infty} 4/(2n)^2$$ etc. etc. A unique form outside of the 1/n^2 family.

Hi, Please see the attached pdf file. Equation 1 and equation 2 are equivalent. Can someone please help me understand how to simplify equation 1 to get to equation 2? Thanks.

Hi I have a textbook which uses the notation: \sum\limits_{i\neq j}^N a_i a_j I can't find anywhere what this actually means. Is it equivalent to: \sum\limits_{i}^N \sum\limits_{j}^N a_i a_j where j can't equal i? Thanks.