Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.

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5
votes
2answers
108 views

Solve the following summation

$S = \dfrac{n \choose 0}{1} + \dfrac{n \choose 1}{2} + \dfrac{n \choose 2}{3}+\dotsb+\dfrac{n \choose n}{n+1}$
0
votes
1answer
15 views

Show $\sum_{d\mid n}\sum_{e\mid(n/d)}\mu(d)f(e))=\sum_{e\mid n}\sum_{d\mid(n/e)}\mu(d)f(e)$

Need verification as to how $$\sum_{d\mid n}\sum_{e\mid(n/d)}\mu(d)f(e))=\sum_{e\mid n}\sum_{d\mid(n/e)}\mu(d)f(e)$$ I am a little unclear as to how this change works, if it could be clarified for ...
1
vote
1answer
72 views

How do I use the principle of mathematical induction to prove whether or not $\sum_{k=1}^n (-1)^k = \frac{(-1)^n-1}2$ is a true statement?

For all n elements of Natural Numbers,$\sum_{k=1}^n (-1)^k= \frac{(-1)^n-1}2$. I proved p(1) to be true : $\sum_{k=1}^1 (-1)^k = (-1)^1 =-1$. And $\frac{(-1)^1-1}2 = \frac{(-2)}2 = -1$ So P(1) ...
2
votes
2answers
40 views

Summation with Riemann Zeta Function

So the Riemann zeta function $\zeta(s)$ is commonly defined as $\sum \limits_{n=1}^{\infty} n^{-s}$ Now, suppose that $a_k=\zeta (2k).$ How can I find the value of $$\sum \limits_{k=1}^{\infty ...
1
vote
1answer
36 views

How do you understand renaming of summation variables?

As a part of a Knuth example, I struggle to understand how you flip the index so easily: $$\sum_{0 < j < k}(k-j) = \sum_{0 < k-j < k} j.$$ Why doesn't Knuth exchange the summand with the ...
0
votes
1answer
14 views

Suppose that number of mistakes on a page is a Poisson RV and independent. From $n$ pages, find the expected number with no mistakes?

A textbook has $n$ pages. The number of mistakes on each page is a Poisson RV with parameter $\lambda$ and is independent of the number of mistakes on all other pages. What is the expected number of ...
1
vote
1answer
36 views

Can $s\sum_{n=0}^{y}(t/s)^{n/y} \ge x$ be solved for $y$?

Is it possible to solve the following equation for y? $s\sum_{n=0}^{y}(t/s)^{n/y} \ge x$ I'm trying to write a slot machine program (for a school assignment I'm making harder than it needs to be for ...
3
votes
1answer
56 views

What is the maximum value of the sum $\sum_{i=1}^L(\bar{x}-x_i)$, in this specific case.

Let $x_i$ be a positive real variable, with $i=1,2,...,K$. We denote by $\bar{x}$ the average value of the values $x_1, x_2,...,x_K$. Let $a=\min_i x_i$ and $b=\max_i x_i$, then $x_i \in [a,b]$. My ...
1
vote
1answer
40 views

Sum of order statistics

Is there a general expression for the pdf of the sum of the order statistics? Suppose they are all drawn independently from the same distribution.
4
votes
2answers
249 views

Limit of an expression

$$\lim\limits_{n\to\infty}\frac{1}{e^n\sqrt{n}}\sum\limits_{k=0}^{\infty}\frac{n^k}{k!}|n-k|=\sqrt{2/\pi}$$ Is this limit true? I should show limit is true. It is allowed to use computer programs to ...
4
votes
2answers
79 views

How to show $c_n=\frac11 + \frac12 + \cdots + \frac1n - \ln n$ is a sequence of positive numbers? [duplicate]

For $n \in \mathbb{N}$ let $c_{n}$ be defined by $$c_{n}=\frac{1}{1} + \frac{1}{2} + \cdots + \frac{1}{n} - \ln n$$ We have to prove that $c_{n}$ is a decreasing sequence of positive numbers. ...
1
vote
1answer
120 views

Evaluate $\sum \sum 1/n^k $

I wanted to evaluate the sum: $$ \sum_{n \ge 2} \left(\zeta(n) - 1\right) $$ I rewrote this as: $$ \sum_{n\ge 2} \sum_{k\ge 2} \frac{1}{n^k} $$ I tried exploiting the symmetry but that didn't seem ...
0
votes
0answers
15 views

How can I find condidions when one sum is greater than the other?

I have the following two sums: $$ \sum_{j\neq i} Q_{ij}u_i\int g(z, k) f(z; i, j) \textrm{d}z $$ and $$ \sum_{j\neq i} Q_{ij}u_j\int g(z, k) f(z; i, j) \textrm{d}z, $$ where $k$ is a parameter and the ...
2
votes
3answers
179 views

Number of ways to express a number as the sum of different integers

Given a number $n$, then $P_k(n)$ is the number of ways to express $n$ as the sum of $k$ integers. For example $P_2(6)=7$ $0+6=6$ $1+5=6$ $2+4=6$ $3+3=6$ $4+2=6$ $5+1=6$ $6+0=6$ Now I worked ...
3
votes
5answers
354 views

How to compute series

I have to compute the series $\displaystyle\sum_{n=0}^{\infty}{\frac{3^n(n + \frac{1}{2})}{n!}}$. $$\displaystyle\sum_{n=0}^{\infty}{\frac{3^n(n + \frac{1}{2})}{n!}} = ...
-1
votes
1answer
29 views

Closed form of sum

How to summarize \begin{equation*} f(x,y)=\sum\limits_{n=0}^{\infty}\frac{\exp\left(-\frac{(2n+1)\pi y}{d}\right)}{2n+1}\sin\frac{(2n+1)\pi x}{d}? \end{equation*} I tried to calculate $\partial_y f$ ...
1
vote
3answers
84 views

Closed form of a sum of ratios of integers

I am computing in a program this sum (does it have a "name"): $$\sum_{\alpha=2}^{K} \frac{\alpha-1}{\alpha}$$ is there a way to avoid the sum, term by term, and use a more compact closed form ?
2
votes
0answers
35 views

Coin tossing heads-counting game probabilities

Suppose I have a coin for which the probability of heads is $p$. Now suppose I have a two-player game where the first player may flip this coin $m$ times and the second player may flip this coin $n$ ...
5
votes
0answers
165 views

Summation of a function 2

Let $n$ is a positive integer. $n = p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is the complete prime factorization of $n$. Let me define a function $f(n)$ $f(n) = p_1^{c_1}p_2^{c_2}\cdots p_k^{c_k}$ where ...
1
vote
2answers
32 views

Suppose $X$ and $Y$ are independent Poisson random variables. Find the conditional probability mass function $P(X=k\mid X+Y=m)$ [duplicate]

Suppose $X$ and $Y$ are independent Poisson random variables with parameters $\lambda$ and $\mu$, respectively. Find the conditional probability mass function $P(X=k\mid X+Y=n)$. Don't know how to ...
8
votes
3answers
201 views

$\sum\limits_{n=1}^{10000000000000000} \frac{1}{n}$

How does wolfram alpha solve $$\sum\limits_{n=1}^{10000000000000000} \frac{1}{n}\approx 37.4186$$so quickly? It solved it in like 3 seconds is there a equation or something
24
votes
11answers
2k views

Is there any way to define arithmetical multiplication as other thing than repeated addition?

Is there any way to define arithmetical multiplication as other thing than repeated addition? For example, how could you define $a\cdot b$ as other thing than $\underbrace{a+a+\cdots+a}_{b ...
1
vote
1answer
27 views

How to solve this kind of Lagrangian function?

Suppose $\mathbf{a} = (a_{0}, \dots, a_{N-1})$ and $\mathbf{b} = (b_{0}, \dots, b_{N-1})$ with $a_{i}\geq0$, $b_{i}\geq 0$. I would like to minimize $$-\sum_{i=0}^{N-1}a_{i}b_{i}$$ subject to ...
1
vote
1answer
34 views

Find a sequence $a_n$ such that its sum converges, but the sum of its logarithm doesn't - Generalisation Help

This question came up in a past paper that I was doing, but it seems to be a fairly common, standard question. Give an example of a sequence $a_n$ such that $\sum(a_n)$ converges, but ...
1
vote
1answer
15 views

Hypothetical Church Growth Equation

To preface this, it's been years since my last calc class, but I feel like this is lower level than that. I am looking for the notation for an equation that sums up the values of the same equation ...
1
vote
1answer
28 views

rational numbers as upper limit of a summation?

a quick question: Is it a legit way to use a fraction as the upper limit of a summation? Given is a frequency $f$ and a sample rate $f_s$. I want to use a sum like this: $\sum_{k=1}^{\frac{f_s}{2f}} ...
1
vote
1answer
34 views

Equivalence between sum expression and power expression

This is a bit silly, but how can I show that $$ \frac{1}{4}\left(3^{x+1}+5\right) = \frac{1}{2}\left(3+\sum_{i=0}^x 3^i\right) $$
0
votes
1answer
31 views

How can we find the sums ?

We have the function $$g: [0, 2\pi] \rightarrow \mathbb{R} \\ g(x)=\frac{(x-\pi)^2}{4}, x \in [0, 2\pi]$$ I found that the Fourier series of $g$ is the following: $$g \sim ...
0
votes
1answer
15 views

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ given by $f(x)=e^{\theta x}$. Show that $E[f(X)]=exp(\lambda (e^\theta -1))$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ given by $f(x)=e^{\theta x}$, where $\theta \in \mathbb{R}$ and suppose that $X ~ Po(\lambda)$ for some $\lambda >0$. Show that $E[f(X)]=exp(\lambda ...
2
votes
4answers
40 views

Suppose that $X$ is a discrete random variable taking values in $\{0,1,2,…\}$. Show that $E[X]=\sum^{\infty}_{k=0}{P(X>k)}$

Suppose that $X$ is a discrete random variable taking values in $\{0,1,2,...\}$. Show that $E[X]=\sum^{\infty}_{k=0}{P(X>k)}$ Absolutely lost. From my notes, we define $E[X]$ as follows ...
-2
votes
1answer
55 views

Finite power series [duplicate]

I'm a student and I'm looking for a solution for the following finite power series: $$ \sum_{n=0}^m \frac{1}{n!} x^n $$ By "solution" I meant expansion of the series and finding a closed form ...
7
votes
6answers
119 views

Intuitively understanding $\sum_{i=1}^ni={n+1\choose2}$

It's straightforward to show that $$\sum_{i=1}^ni=\frac{n(n+1)}{2}={n+1\choose2}$$ but intuitively, this is hard to grasp. Should I understand this to be coincidence? Why does the sum of the first ...
0
votes
1answer
62 views

Simplifying a sum of products related to Vandermonde determinant

How to show this equality? $$ 1=(-1)^n\sum_{k=0}^n\frac{x_k^n}{\prod_{\substack{l=0 \\ l \neq k}}^n(x_l-x_k)} $$ This is part of a proof to show the value of the determinant of the Vandermonde matrix ...
-1
votes
1answer
68 views

Finding value of the summation

How can I solve the following summation for a set $S$? $$\sum\limits_{s \subset S} \left[\max(s)-\min(s)\right]$$ Example : If $S = \{1,5,2\}$ Subset $= \{1\}$, $\max(s)-\min(s) = 0$. Subset $= ...
0
votes
1answer
23 views

Derivation of a summation

I understand that $\sum\limits_{t=1}^{\infty}$ $t(1 - p)^{t}$ = $\tfrac{1-p}{p^2}$, for mod(1-p) $<$ 1. However, I am trying to derive this fact and I'm not sure so please could someone help.
-2
votes
4answers
76 views

Evaluate the sum $1+2+3+…+n$

How do we evaluate the sum: \begin{equation*} 1+2+...+n \end{equation*} I don't need the proof with the mathematical induction, but the technique to evaluate this series.
0
votes
2answers
42 views

Proof of a the following formula: $\sum_{k=0}^n \frac{1}{(k+1)(n-k+1)}=\frac{2}{(n+2)}\sum_{k=1}^{n+1} \frac{1}{k}$

How can I prove the following combinatorial identity about the harmonic series? $$\sum_{k=0}^n \frac{1}{(k+1)(n-k+1)}=\frac{1}{(n+2)}\sum_{k=0}^n \left(\frac{1}{(k+1)} + ...
0
votes
1answer
31 views

Difficulty understanding Sums - Number Theory

I've been trying to understand the following equality for quite some time. And since I bump into it frequently I cannot oversee it: $$\underset{d|n}\sum d=\underset{d|n}\sum ...
0
votes
2answers
10 views

Given the integral of an equation over one set of bounds find the integral over another set of bounds.

If $\int_{1}^{3}f(w)dw=7$, find the value of $\int_{1}^{2}f(5-2x)dx=7$ I think this problem has something to do with the fact that (5-2(2)) = 1 and (5-2(1)) = 3 and these are the bound of the ...
3
votes
5answers
56 views

Formula for $r+2r^2+3r^3+…+nr^n$ [duplicate]

Is there a formula to get $r+2r^2+3r^3+\dots+nr^n$ provided that $|r|<1$? This seems like the geometric "sum" $r+r^2+\dots+r^n$ so I guess that we have to use some kind of trick to get it, but I ...
0
votes
0answers
11 views

recurrence tree final step - binary search

Starting with the base case and recursive case run times as follows: 􏰀 t(N) = 1 , if N = 1 t(N)= 1+t(N/2) ,ifN > 1 At the end of my tree I have ...
2
votes
2answers
64 views

How to obtain a closed form for summation over polynomial ($\sum_{x=1}^n x^m$)? [duplicate]

What is the method for obtaining the polynomial equal to \begin{equation*} \sum^{n}_{x=1}x^m \end{equation*} for unknown $n$, and systematically for various values of $m$? I know it should be a ...
0
votes
0answers
27 views

“+” operator placed as index

What is the meaning of $(a-b)_+$? In other words, what is meant by the "+" operator when it is placed as an index. If I am comparing for example two variables $a$ and $b$. So what is the value of ...
0
votes
0answers
36 views

Is this function $2\pi$-periodic?

If I construct $$G(x) =\sum_{1}^{\infty} g(x +2n\pi),$$ does this make $G(x)$ $2\pi$-periodic? My understanding is that if $G(x)$ were now $2\pi$-periodic, then that means $G(x) = G(x + 2\pi$) = G(x ...
2
votes
3answers
54 views

Differentiation method for evaluating $ \sum_{n=1}^\infty \frac{n^2}{3^n} $

I evaluated the following infinite sum (the original and broader question regarding this sum can be found at Evaluating $\sum_{n=1}^\infty \frac{n^2}{3^n} $). $$ \sum_{n=1}^\infty \frac{n^2}{3^n} $$ ...
0
votes
0answers
41 views

How to convert infinite intergral to sum

How to convert Wiener filter formulas from integral to sum? They are for images therefore it must be possible to convert them to sums. Any help will be appreciated: I could not find much info on ...
2
votes
1answer
28 views

Why is $\sum_{x=1}^{p-1}\left(\frac{x}{p}\right)=\left(\frac{0}{p}\right)$?

For $p$ an odd prime, why does $$\sum_{x=1}^{p-1}\left(\frac{x}{p}\right)=\left(\frac{0}{p}\right)$$ where $\left(\frac{x}{p}\right)$ is the Legendre symbol. I'm not sure if I have given enough ...
0
votes
3answers
66 views

What does $\sum\limits_{n=1}^{N-1} \frac{1}{n} - \sum_{n=3}^{N+1} \frac{1}{n} $ simplify to?

A solution to one of the exercises in my text states: $$\sum\limits_{n=1}^{N-1} \frac{1}{n} - \sum_{n=3}^{N+1} \frac{1}{n} = \frac{1}{1} + \frac{1}{2} - \frac{1}{N} - \frac{1}{N+1}$$ I have no idea ...
1
vote
2answers
71 views

Confusion about how to prove $\sum_{i=0}^n 2^i = 2^{n+1}-1$ for all $n\geq 0$ by induction

I'm trying to understanding proof by induction. But how do I check if that is correct? How do I know what I need to show? Any help would be great. Just trying to get my head around this. So I have ...
1
vote
2answers
65 views

What is the sum $\sum_{k=0}^{n-1} e^{kx}$? [closed]

My Precalc teacher gave me this as a question and I simply cannot figure out how to do it.