Articles of summation

Finding sum $\sum_{i=1}^n \frac1{4i^2-1}$

I have been having problem with calculating the following summation: $$ \sum_{i=1}^n {1\over 4i^2-1} = {1\over3} + {1\over15} + {1\over35} + \cdots + {1\over 4n^2-1} $$ I do know the answer, but just can not find the way to get it. Thank you in advance.

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 this function they partially differentiate this function with respect to $x_i$. $$\frac{\partial{snr} }{\partial{x_i}} = \frac{\dfrac{t_i}{d_i}-\frac{\sum_{j=1}^n x_jt_j}{\sum_{j=1}^n x_jd_j}}{\frac1{d_i}\sum_{j=1}^n x_jt_j} $$ any clues as to how to […]

Prove (or disprove) that $ \sum_{n=1}^\infty \frac{4(-1)^n}{1-4n^2} x^n = \frac{2(x+1) \tan^{-1}(\sqrt x)}{\sqrt x} – 2 $ for $ 0<x\leq1$

Just like title said, for $ 0 <x\leq1 $, prove/disprove: $$ \displaystyle \sum_{n=1}^\infty \dfrac{4(-1)^n}{1-4n^2} \cdot x^n \stackrel{?}{=} \dfrac{2(x+1) \tan^{-1}(\sqrt x)}{\sqrt x} – 2 $$ I got this equation from Claude Leibovici. It’s true for $ n=1 $ as shown by Ron Gordon. I think it’s feasible to show that it’s true from the RHS by […]

Verify $y=x^{1/2}Z_{1/3}\left(2x^{3/2}\right)$ is a solution to $y^{\prime\prime}+9xy=0$

This question is a sequel to this previous question. As before, some background information is needed first as follows from my textbook: The standard form of Bessel’s differential equation is $$x^2y^{\prime\prime}+xy^{\prime} + (x^2 – p^2)y=0\tag{1}$$ where $(1)$ has a first solution given by $$\fbox{$J_p(x)=\sum_{n=0}^\infty\frac{(-1)^n}{\Gamma(n+1)\Gamma(n+1+p)}\left(\frac{x}{2}\right)^{2n+p}$}\tag{2}$$ and a second solution given by $$\fbox{$J_{-p}(x)=\sum_{n=0}^\infty\frac{(-1)^n}{\Gamma(n+1)\Gamma(n+1-p)}\left(\frac{x}{2}\right)^{2n-p}$}\tag{3}$$ where $J_p(x)$ is called […]

modulo of series summation

I have trouble with computing modulo. First, I have a summation of series like this: $$1+3^2+3^4+\cdots+3^n$$ And this is the formula which can be used to compute the series: $$S=\frac{1-3^{n+2}}{1-3^2}=\frac{3^{n+2}-1}{8}$$ Then I want to compute $S \mod 1000000007$. But if $n$ is a large number, it’s really hard for me to compute it. The only […]

Multi-index power series

What is closed-form expression for the summation $$ S(n,m)=\sum_{|\alpha|=m} p^{\alpha} = \sum_{\alpha_1 + \cdots + \alpha_n = m} \prod_{i=1}^n p_i^{\alpha_i} $$ as a function of $n$ and $m$? Here $\alpha=(\alpha_1,\cdots,\alpha_n)$ is a $n$-tuple of non-negative integers. For specific values of $n$ the sum has a closed form, e.g. $S(1,m)=p_1^m, S(2,m)=\frac{p_2^{m+1}-p_1^{m+1}}{p_2-p_1}$, etc. I wonder if there […]

Algebra and Substitution in Quadratic Form―Einstein Summation Notation

Schaum’s Outline to Tensor Calculus ― chapter 1, example 1.5 ――― If $y_i = a_{ij}x_j$, express the quadratic form $Q = g_{ij}y_iy_j$ in terms of the $x$-variables. Solution: I can’t substitute $y_i$ directly because it contains $j$ and there’s already a $j$ in the given quadratic form. So $y_i = a_{i \huge{j}}x_{\huge{j}} = a_{i \huge{r}}x_{\huge{r}}$. […]

Relative error of machine summation

Let $\mathbb{F}(b,t,L,U)$ (briefly $\mathbb{F}$) the set of all machine numbers. The definition is the usual, i.e. $\mathbb{F}$ is defined as \begin{equation*} \mathbb{F} := \left\lbrace (-1)^{s} m b^{e-t}\right\rbrace \end{equation*} where $b^{t-1} \leq m \leq b^{t}-1$, $L \leq e \leq U$ and $s \in \left\lbrace 0,1\right\rbrace$. The lower bound imposed on $m$ guarantees the uniqueness of the […]

Prove that $\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3} +…+ \frac{n}{2^n} = 2 – \frac{n+2}{2^n} $

I need help with this exercise from the book What is mathematics? An Elementary Approach to Ideas and Methods. Basically I need to proove: $$\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3} +…+ \frac{n}{2^n} = 2 – \frac{n+2}{2^n} $$ $i)$ Particular cases $ Q(1) = \frac{1}{2} ✓ $ ———- $ P(1) = 2 – \frac{1+2}{2^1} = \frac{1}{2} ✓ $ $+ = 1 […]

How to calculate the sumation of a function in one step?

I’m building a Hierarchical Agglomerative Clustering algorithm and I’m trying to estimate the time the computer will take to build a hierarchy of clusters for a given set of samples. For $m$ samples, I have to calculate $m-1$ levels in a binary dendrogram. For each level with $n$ elements, I have to calculate $\sum_i^ni-1=\frac{(n-1)*n}{2}$ distances. […]