0
votes
0answers
20 views

Summation of a function with the variable both in the function amd in the upper limit

E is defined as : E = c1 ( a$\rho$ + b$\rho ^{2}$ ) + c2 $\rho$ ( c + d $\sum_{j=0}^{n} (\log{ \frac{R\rho}{j} } ) $ ) + c3 $\rho ^{2}$ a, b, c, d, c1, c2, c3, R are known constants. $\rho$ is the ...
1
vote
0answers
24 views

Prove the partial derivative of the summation of $$(y-g_i+a\sum^n_{j=1} g_j)=-1+na >0$$

I have a function: $$\pi_i^1=y-g_i+a\sum^n_{j=1}g_j,$$ where 0 < a<1< na, and I need to prove this: $$\frac{\partial(\sum^n_{i=1}\pi^1_i)}{\partial g_i}=-1+na>0.$$ I am not very ...
2
votes
1answer
118 views

Derivative of double summation and dot notation?

I am trying to differentiate the following summation: $$ L(\mu, \tau_1, \ldots, \tau_i)= \sum_{i=1}^v \sum_{t=1}^{r_i} (y_{it}-\mu - \tau_i)^2 $$ $$\frac{dL}{d\mu} = y_{\cdot\cdot}-n\mu - ...
0
votes
1answer
18 views

Make derivations over sums

I have this kind of sums $$ \left(\sum_{i_1=0}^{4}\sum_{i_2=0}^{4}\log(f(X,i_1,i_2))\right)'\ $$ And we want to derive in respect to $${x_i}$$, which is an element of the vector X. How I should do ...
5
votes
1answer
101 views

Simplify $\sum_{n=0}^{N}\binom{N}{n} \frac{a^{N-n}}{n!} \frac{d^n}{dx^n} f(x)$

Simplify the following expression $$S_N = \sum_{n=0}^{N}\binom{N}{n} \frac{a^{N-n}}{n!} \frac{d^n}{dx^n} f(x), $$ where $a$ is a real number and $f(x)$ is an analytic real function. What is $\lim_n ...
3
votes
3answers
132 views

Find the 100th derivative of $x \sinh(2x)$

If $f(x) = x \sinh(2x)$, find $f^{({100})}(x)$. My (Incorrect) working so far: Using Leibniz' Formula for derivatives: $$(fg)^{(n)}=\sum_{k=0}^n{n\choose k}f^{(k)}g^{(n-k)}$$ ...
0
votes
2answers
110 views

derivative of a summation with variable upper limit

Is the following statement correct and if yes does it need to satisfy specific requirement to be correct: $${ d \over dt} \sum_{j=1}^{N(t)} f(t,j) = \sum_{j=1}^{N(t)} {df(t,j) \over dt} + f(t,N(t)) ...
1
vote
1answer
39 views

Derivative of a summation

I need to compute the derivative of this function: $f(\alpha) = \sum_{i=1}^n \left[U_i - U_0 \left( \frac{h_i}{h_0} \right)^\alpha \right]^2$ where $h_0$ and $U_0$ are constant. I thought it was ...
2
votes
0answers
56 views

Why does $\frac{d}{d\theta} \theta\prod_{i=1}^nx_i = \sum_{i=1}^nx_i$

Is this just the product rule? I have this in my notes but I didn't think anything of it and now I'm wondering how this happens? Edit: Im working with maximum likelihood estimation and in my notes I ...
0
votes
3answers
179 views

Derivative of a summation in order to minimize

I am asked to minimize $\sum^n_{i=0}(x_i - C)^2$ with respect only to C so I know I have to take the derivative respect to C, set it equal to 0, and then solve. I have never done summation in my ...
1
vote
1answer
62 views

Why does this interchanging of derivative and sum work?

I'm reading a stats book and, for a geometric distrubution ($E[Y]=p \sum_{y=1}^{\infty}yq^{y-1})$ it makes the claim that since $\displaystyle \frac{d}{dq}(q^y)=yq^{y-1}$ hence $\displaystyle ...
2
votes
1answer
65 views

complicated derivative with nested summations

How would I solve for this derivative? $$s=\frac{1}{N} \sum_i^N \left[t_i - \left(\sum_j \left[c_j e^{-\frac{(r_i-r_j)^2}{2w^2}} + b\right]\right)\right]^2$$ I want to solve for $\dfrac{ds}{dw_j}$. ...
0
votes
1answer
62 views

Binomial sum of derivatives

I would like to know the result of the following sum: $$\sum_{p=0}^m \binom{m}{p}(-1)^{p-1}\frac{\partial^{p-1}}{\partial x^{p-1}}f(x)\cdot(-1)^{m-p-1}\frac{\partial^{m-p-1}}{\partial ...
0
votes
0answers
73 views

Second derivative of infinite sum

Is the following function: $$f(x)=\sum_{n=1}^\infty {\frac{\sin(nx^2)}{1+n^4}}$$ Has a continuous second derivative in R? (If it was in $[-a,a]$ or for $\sin(nx)$ it was easy, but I'm not sure what ...
1
vote
2answers
957 views

Differentiation of summation of summation

According to http://www.atmos.washington.edu/~dennis/MatrixCalculus.pdf, (45) and (46) (p. 6), differention of $$\alpha = \sum_{j=1}^n\sum_{i=1}^n a_{ij} x_i x_j $$ with respect to the k-th element ...
0
votes
0answers
124 views

Derivative of a sum with respect to the maximum step?

I am looking to take the derivative of $\partial _n(\sum_ {j=n/2}^{n} f(n,j))$ and I am not sure how to go about doing it. Can Anyone point Me in the direction of information about how to proceed? ...
4
votes
3answers
744 views

Solve $\sum nx^n$

I am trying to find a closed form solution for $\sum_{n\ge0} nx^n\text{, where }\lvert x \rvert<1$. This solution makes sense to me: $\sum_{n\ge0} x^n=(1-x)^{-1} \\ \frac{d}{d x} \sum_{n\ge0} x^n ...
2
votes
1answer
42 views

How do I calculate the sum of $\sum_{k=1}^{\infty}\frac{(2-x)^k}{2^k\cdot k}$ in every x in (0, 4)?

Well I've been trying to search for the appropriate derivative but I couldn't find it Thanks
1
vote
1answer
31 views

Help manipulating a sum

Let $S(x,t)=e^{2xt-t^2}=\sum^\infty_{n=0}\frac{H_n(x)}{n!}t^n$ If we now differentiate each term with respect to $x$ we find: \begin{align*} \frac{\partial S}{\partial ...
2
votes
1answer
875 views

Partial derivative of a summation.

I am trying to confirm a stated result on my lecture slide. Question: Given that $A:= \sum_i^n \frac{a_i}{(1+b)^{t_i}}$, where $a_i,b \in \mathbb{R}_+$ and $t_i \in \{t_1,...,t_n\}$ where $0 < ...
4
votes
1answer
190 views

Computing a finite binomial sum

I want to compute $$S(n,m,a)=\sum_{k=0}^{n}k^{m}\cdot\binom{n}{k}\cdot a^k.$$ With $n,m\in\mathbb N$, $a\neq0$ and $S(n,0,a)=(a+1)^n$. What I have found already: I don't see any other options then ...
1
vote
2answers
1k views

differentiation with summation symbol

I am trying to understand a step in the math given a scientific paper. They differentiate an objective function of the form: $$snr = \frac{\sum_{i=1}^n x_it_i}{\sum_{i=1}^n x_id_i} $$ To maximize ...