Infinite sums Definition and 18 Threads

I have a sum that looks like the following: ## \sum_{k = 0}^{\infty} \left( \frac{A}{A + k} \right)^{\eta} \frac{z^k}{k!} ## Here, A is positive real. If \eta is an integer, this can be written as: ## \sum_{k = 0}^{\infty} \left( \frac{A(A +1)(A+2) \cdots (A + k - 1)}{(A + 1)(A+2)(A+3)...

TL;DR Summary: I want to find a function with f'>0, f''<0 and takes the values 2, 2^2, 2^3, 2^4,..., 2^n Hello everyone. A professor explained the St. Petersburgh paradox in class and the concept of utility function U used to explain why someone won't play a betting game with an infinite...

I have a doubt about the notation and alternative ways to represent the terms involved in sums. Suppose that we have the following multivariable function, $$f(x,y)=\sum^{m}_{j=0}y^{j}\sum^{j-m}_{i=0}x^{i+j}$$. Now, let ##\psi_{j}(x)=\sum^{j-m}_{i=0}x^{i+j}##. In the light of the foregoing, is...

Hello guys, I struggle with one step in a calculation to show a quantum operator equality .It would be nice to get some help from you.The problematic step is red marked.I make a photo of my whiteboard activities.The main problem is the step where two infinite sums pops although I work...

Hi. I know that eixe-ix = 1 but if I write the product of the 2 exponentials as infinite series I get ΣnΣm xn/(n!) (-x)m/(m!) without knowing the result is 1 using exponentials how would I get the result of this product of 2 infinite sums ? Thanks

The problem I'd like to calculate the value of this sum: $$3 \sum^\infty_{k=1}\frac{1}{2k^2-k}$$The attempt ## 3 \sum^\infty_{k=1}\frac{1}{2k^2-k} = [k=t/2] = 3 \sum^\infty_{t=2}\frac{1}{2 \left( \frac{t}{2} \right)^2-\frac{t}{2}} = 3 \sum^\infty_{t=2}\frac{1}{ \frac{t^2}{2} - \frac{t}{2}} = 3...

Homework Statement Modify the initial conditions (for the diffusion equation of a circle) to have the initial conditions ## g(\theta)= \sum_{n=-\infty}^{\infty}d_{n}e^{2\pi in\theta} ## Using the method of Green's functions, and ## S(\theta,t)= \frac{1}{\sqrt{4\pi...

Hello there. I'm here to request help with mathematics in respect to a problem of quantum physics. Consider the following function $$ f(\theta) = \sum_{l=0}^{\infty}(2l+1)a_l P_l(cos\theta) , $$ where ##f(\theta)## is a complex function ##P_l(cos\theta)## is the l-th Legendre polynomial and...

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...

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...

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}...

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...

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!

I'm working on a calculus project and I can't seem to work through this next part... I need to substitute equation (2) into equation (1): (1): r\frac{\partial}{\partial r}(r\frac{\partial T}{\partial r})+\frac{\partial ^{2}T}{\partial\Theta^{2}}=0 (2): \frac{T-T_{0}}{T_{0}}=A_{0}+\sum from n=1...

i am trying to re-express the following in terms of a rational function: \frac{(0+x+2x^2+3x^3+...)}{1+x+x^2+x^3+...} . i know that this is supposed to be \frac{1}{x-1} but I can't figure out how to do it. I know the denominator is just \frac{1}{1-x}. so in order for this work out, the...

I understand that the limit of the sum of two sequences equals the sum of the sequences' limis: \displaysyle \lim_{n\rightarrow\infty} (a_{n} + b_{n}) = \lim_{n\rightarrow\infty}a_{n} + \lim_{n\rightarrow\infty}b_{n}. Similar results consequenly hold for sums of three sequences, four sequences...

Homework Statement Using the macluarin's expansion for sinx show that \int sinx dx=-cosx+cHomework Equations sinx=\sum_{n=0} ^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!} The Attempt at a Solution Well I can easily write out some of the series and just show that it is equal to -cosx but if I...