Sum Definition and 1000 Threads

Can someone please show me how the formula [SIZE="4"]\sum^{∞}_{0}\frac{u^{x}}{x!} = e^{u} Is derived? Or link me to an explanation. Thanks! http://www.wolframalpha.com/input/?i=sum+of+%28%28a%5Ex%29%2F%28%28x%29%21%29%29+from+x%3D0+to+x+%3D+inf (Just to show you what I'm talking about)

Here is the question: Here is a link to the question: (CALCULUS)The sum of two numbers is k. Show that the sum of their squares is AT LEAST (1/2)(k^2)? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

I asked the following question on facebook Prove that $$\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4} \, \cdot\cdot \cdot \,+ \frac{1}{n(n+1)}=\frac{n}{n+1} $$

Sum from 0 to infinity of [3/[(n+1)(5^n)]](x-3)^n HELP :) 1. ∞ Ʃ \frac{3}{(n+1)(5^{n}}*(x-3)n n=0 2. The first question I had to answer was: What is f(3)? I found the first 4 terms to be: 3, 3/10(x-3), 3/75(x-3)2, 3/189(x-3)3 So f(3) equals 3, I'm pretty sure. Because...

B(t) is a standard Brownian Motion. u and v are both => 0. What is the distribution of B(u) + B(v)? The mean is 0. For the variance I get Var(B(u)+B(v)) = u+v. Is this right?

1. ∞ Ʃ (1+n)/[(n)(2n)] n=1 2. When I see that there is an n as an exponent, I think to do the ratio test. ___________________________________________________________________________ \frac{\frac{1+n+1}{(n+1)(2^{n+1})}}{\frac{1+n}{(n)(2^{n})}} = \frac{n(n+2)}{2(n+1)^{2}} =...

1. ∞ Ʃ √(n)/(n2 + 1) n=1 Find if it converges. 2. I'm wondering if I can rewrite this by bringing the n1/2 down to the denominator, making it negative... 1/(n-1/2)(n2 + 1) = 1/(n-1 + n-1/2) = n + √(n) ...And it seems to me that this one would diverge because the n value...

I'm really confused about the double sum given by my textbook. Here's what it says: If A is an nxm matrix, its length is the square root of the sum of its squares of all its entries. \left|A\right|^{2}=\sum^{n}_{i=1}\sum^{m}_{j=1}a_{i,j}^{2} The double sum is what has me caught up. How...

Question: Given a non-negative integer N, show many sets of non-negative integers (a,b,c,d) satisfy 2a+b+c+d=N Proposed (and roadblocked) solution: Case 1: 2a=0 Then there are \binom{N+2}{2} solutions (easy to prove). Case 2: 2a=2 Then there are \binom{N+2-2}{2} solutions. Case 3: 2a=4...

I'm not really sure when each of these should be done. In fact, I don't really understand the reason that we use the limit comparison test. Σ1/(n^2+1) So here I can simply say that P=2>1, so the original converges. Σ1/N^3+N^2 Here, I would say that P=3>1, implying the original...

1. Find the sum of the convergent series: ∞ Ʃ 2/[(4n-3)(4n+1)] n=12. Hm... Okay, so I started with the nth term test, and the denominator gets huge very fast. So I'm pretty sure it goes to zero. So that tells us nothing other than that it does not FOR SURE diverge. Since it has no n in the...

Homework Statement Use partial fractions to find the sum of the series.Homework Equations \displaystyle \sum^{∞}_{n=1} \frac{8}{n(n+3)} The Attempt at a Solution I end up with: \displaystyle \frac{8}{3n} - \frac{8}{3(n+3)} I am stuck here.

If $\displaystyle y=\frac{3}{4}+\frac{3*5}{4*8}+\frac{3*5*7}{4*8*12}+...\infty$. Then $y^2+2y = $

1. The problem statement Find the sum of the series: a. 1 + a cos θ + a^{2} cos 2θ + a^{3} cos 3θ + ... + a^{n} cos nθ Apparently, the answer is: \frac{a^{n+1}(a cos nθ - cos(n+1)θ) - a cos θ + 1)}{a^{2} - 2a cos θ + 1} 2. The attempt at a solution = The real part of z^{0} +...

Homework Statement Hello, My homework is online it would be to much trouble for me to type it all out it has a diagram with it. I cannot copy and paste it either. So if someone would be so kind as to message me I can send it through email as a word doc with a screen shot attached. I have...

Homework Statement Determine whether the series is convergent or divergent. If it is convergent, find its sum. \sum\limits_{n=1}^{\infty} (\frac{1}{e^n}+\frac{1}{n(n+1)}) Homework Equations The Attempt at a Solution So I found it's convergent: \sum\limits_{n=1}^{\infty}...

Hello, I am asked to evaluate the following expression using functions meshgrid, sum and dot operations in MATLAB: y = Ʃ(n=1 to N) xn*[cos(x2 + n2)/xn], where x is the vector of four equally spaced values from 1 to 2, N=10. Below is my attempt (I am quite positive it's incorrect, though): n =...

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 ...?

Is there any general expression for the sum to nth term of the series 1/k? I know that for sufficiently large indices, a good approximation can be ln(b/a) where b and a are the upper and lower limits respectively. I've tried to do something very simple for the exact sum by constructing...

If σ(N) is the sum of all the divisors of N and τ(N) is the number of divisors of N then what is the sum of sum of all the divisors of first N natural numbers and the sum of the number of divisors of first N natural numbers? Is there any relation between σ(N) and τ(N) functions? Can I do that...

Given a large number N, do we have any formula to find the number of prime factors and their sum like τ(N) and σ(N) functions? CONDITION: One should not list the factors of N or is not allowed to factorize N since afterwards it would be just a matter of counting and addition

Homework Statement This is a two part question, though once one is solved the other should be the same process: "Write z=re^(it), where 0 < r < 1, in the summation formula and then with the aid of the theorem show that \sum r^n*cos(nt) = (r cos (t) - r^2)/(1-2r*cos(t) + r^2) when 0 < r < 1...

Hi everyone, I've been looking for the finite sum formulae of trig functions. I've found the easiest ones (sine and cosine). But the one for the tangent seems to be very hard. No mathematical tricks work. Plus I've looked it up on the internet. Nothing. I will greatly appreciate your help...

I think I'm missing something really simple here due to being out of school for a while: In a free body diagram, if I am trying to take the sum of torques about point 1, how should I deal with an applied torque at point 2? See attached sketch

Homework Statement f(x)=x^{2} from [a,b] Prove that F(x)=\frac{b^{3}}{2}-\frac{a^{3}}{2}Homework Equations The Attempt at a Solution Using the definition of an integral, we get: U(f,P)=\sum^{n}_{i=1}M_{i}(t_{i}-t_{i-1}) L(f,P)\sum^{n}_{i=1}m_{i}(t_{i}-t_{i-1}) for the function x2, how do we...

Hey, it is clear for me that \sum_{i=1}^{n} i = \frac{n(n+1)}{2} how to find a formula for \sum_{i=1}^{n} i^2 \sum_{i=1}^{n} i^3 Thanks

Let W1 = {A\in MnXn(R)| A = AT} and W2 = {A\in MnXn(R)| A = -AT} Show that MnXn = W1 (+) W2 where the definition of direct sum is: V is the direct sum of W1 and W2 in symbols: V = W1 (+) W2 if: V = W1 + W2 and W1 \cap W2 = {0} Attempt: I figure I have to show each...

Prove that if $\displaystyle \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=1$, then $\displaystyle \frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}=0$

Hi all, Say that I already know W1, W2 are both subspaces of a vector space V, W1∩W2={0}, and that dim(W1)+dim(W2)=dim(V)=n, can I thus conclude that V=W1+W2, namely V is the direct sum of W1 and W2?

Homework Statement As part of a problem I have to show that lim_{n\to\infty}\sum_{i=\frac{n}{2}}^{n}\frac{1}{i}=ln(2) Homework Equations Taylor expansion of ln(2): \sum_{i=1}^{\infty}\frac{(-1)^{k+1}}{k} The Attempt at a Solution ln(2) can be written as: ln(2) =...

Homework Statement Ʃ(x-->infinity, x>0) 11/(n(n+2)) Homework Equations The Attempt at a Solution I am not quite sure what to do, i think that i am supposed to put it into partial fractions. i changed it to the form 11/(n^2+2n) --> 11/((n+1)(n+1)-1)) --> 11/((n+1)^2-1)...

(1/1-1/6) + (1/2-1/7) + (1/3-1/8)...+(1/95-1/100) This was on the cover of a local paper with the caption "You can't solve this but he can!" The 'he' in the caption was a 12 year old boy. On the next page they gave as his solution 2.2. Two things are clear, 1.-the boy had the right answer and...

My question arises in the context of bosonic string theory … calculating the number of dimensions, consistent with Lorentz invariance, one finds a factor that is an infinite sum of mode numbers, i.e. positive integers … but it really goes back to Euler, and his argument that the sum of all...

I am curious if there is a universal formula to find all possible sums of a given number. For instance, to add to 10: 1+9 2+8 1+1+8 2+2+2+3+1, etc I came up with a simple algorithm, but I'm sure there is something similar to Gauss's formula which can be utilized. I have heard Partitions...

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...

Hi all, Can anyone gimme any clues to solve the sum below (or solve it outright :D)? \sum_{i=k}^{n} \frac{i!}{(i-k)!} I'm trying to solve one of Feyman's logic problems (bored geek alert) and I'm stuck at this point. And since my high school days are so far behind... Thanks in...

Homework Statement Find a value for n for which the nth partial sum is ensured to approximate the sum of the alternating harmonic infinite series to three decimal places. Homework Equations Sn = Ʃ(-1)^k+1*1/k = 1 - 1/2 + 1/3 - 1/4 + 1/5 - . . . S1 = 1 S2 = 1 - 1/2 S3 = 1 - 1/2 + 1/3 S4...

Homework Statement I just have to prove the well known identities: \cos(\alpha + \beta)=\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta) \sin(\alpha + \beta)=\sin(\alpha)\cos(\beta)+\cos(\alpha)\sin( \beta) But the thing is that I've to use the Taylor power series for the sine and cosine...

Hello everybody, I'm new to this forum so thanks for having me. I'm trying to find the times when the extrema occur for a periodic wave f(t) equal to the sum of three sine waves. Given f(t) = sin(2∏at) + sin(2∏bt) +sin(2∏ct) where a, b and c are whole numbers in lowest form (i.e...

I always see problems like "how many structurally distinct abelian groups of order (some large number) are there? I understand how we apply the theorem which tells us that every finite abelian group of order n is isomorphic to the direct sum of cyclic groups. We find this by looking at the...

Homework Statement Let a_{n} be an alternating series whose terms are decreasing in magnitude. How to compute the sum as precisely as possible using four-digit chopping arithmetic? In particular, apply the method to compute \sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}}}{{(2n)!}}} and...

How to prove the convergence or divergence of ? $$\sum^{\infty}_{n=1}\frac{\Gamma(n+\frac{1}{2})}{n\Gamma{(n+\frac{1}{4})}}$$

[FONT=arial]Problem: Let A, B, C be three non-collinear points. Let D, E, F be points on the respective interiors of segments BC, AC and AB. Let θ, φ and ψ be the measures of the respective angles ∠BFC, ∠CDA and ∠AEB. Prove IAS(ABC) < θ +φ + ψ < 540 - IAS(ABC).(IAS means internal angle sum). Now...

Hi members of the forum, Problem: Rationalize the denominator of $\displaystyle \frac{1}{a^\frac{1}{3}+b^{\frac{1}{3}}+c^{\frac{1}{3}}}.$ I know that if we are asked to rationalize, say, something like $\displaystyle \frac{1}{1+2^{\frac{1}{3}}}$, what we could do is the following...

Suppose we have two multivariate functions, u_{1}(x,t) and u_{2}(x,t). These functions are solutions to second-order linear equations, which can be written as follows: Au_{xx}+Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}+Fu=G Each of the coefficients are of the form A(x,y). Now, the linearity of these...

I'd really appreciate some help with a sum of: a_n= |sin n| / n All I've thought of, is that I should probably create a subsequence of {a_n}, such that all the elements of this subsequence {a_n_k} are >epsilon >0, and then compare the subsequence to 1/n which diverges. However, I have no...

Homework Statement Find the sum of 5^1-5^2+5^3-5^4+...-5^{98} a. (5/4)(1-5^99) b. (1/6)(1-5^99) c. (6/5)(1+5^98) d. (1-5^100) e. (5/6)(1-5^98) Homework Equations The Attempt at a Solution I feel as though this is actually a simple problem and that I'm not looking at it the right way. [5^1...

the answer is 3^2048. How do I get there?

Homework Statement The Attempt at a Solution Alright, so, I'm clueless about doing this one. I do know that it's extremely similar to e^x \sum_{n->0}^{\infty}\frac{x^n}{n!} But really, that means nothing! Usually there's another function/series I can compare and then integrate, in...

If I have a periodic function that is say a sum of a number of sine functions I can use a Fourier Transform to get the component functions. Now, if I have a bunch of square waves of differing amplitudes and frequencies that I add up into a resultant waveform. Given that waveform what'd be...