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

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5
votes
5answers
121 views

Prove that $\sum_{k=0}^n \binom{3n-k}{2n}=\binom{3n+1}{n}$

Prove that $$\sum_{k=0}^n \binom{3n-k}{2n}=\binom{3n+1}{n}$$ I've tried multiple things that didn't work. Maybe this would help $$\sum_{k=0}^n \binom{3n-k}{2n}=\sum_{k=0}^n \binom{3n-(n-k)}{2n}=\...
5
votes
0answers
65 views

How many egyptian fractions including and above (1/n) are necessary to sum to unity

Let there be a finite set of positive integers such that: (a) no two members of the set are equal (b) the sum of the inverse of each member of the set is equal to one The smallest set (as defined ...
1
vote
2answers
108 views

Combinatorial proof of $\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}(l-k)^n=n!$, using inclusion-exclusion

If $l$ and $n$ are any positive integers, is there a proof of the identity $$\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}(l-k)^n=n!\;$$ which uses the Inclusion-Exclusion Principle? (If necessary, ...
4
votes
2answers
48 views

Is it possible to put series in other series?

I've been working on a project for quite a long time but I found myself stuck at a step where I have to multiply elements of a series by elements of another series, which is dependent on the former ...
1
vote
0answers
41 views

Show that $\sum (-1)^n x^{(2^n)}$ has no limit as $x \uparrow 1$

Show that the following limit does not exist: $$\sum_{0}^{\infty} (-1)^n x^{(2^n)}\text{ with }x \uparrow 1$$ I tried setting $$f(x) = x - x^2 + x^4 - x^8...$$ then $$f(x) = x-f(x^2)$$ then the ...
0
votes
2answers
50 views

Finding the sum of $\cos45°$ + $i\cos135°$ + … + $i^{n}\cos(45+90n)°$ + … + $i^{40}\cos3645°$

My question is as follows: If $i^{2}$ = -1, find the value of $$\cos45° + i\cos135° + \ ...\ + i^{n}\cos(45+90n)° + \ ...\ + i^{40}\cos3645°$$ without the aid of a calculator. In terms of my attempts ...
2
votes
2answers
71 views

Show that $\sum_{k=0}^n \frac{(2n)!}{{k!^2(n-k)!}^2}= \binom{2n}{n}^2$

Show that $$\sum_{k=0}^n \frac{(2n)!}{k!^2(n-k)!^2} = \binom{2n}{n}^2.$$ I tried canceling $2n!$ from both sides then moving $k!$ to right but still not sure how to proceed.
6
votes
4answers
166 views

Proof of the summation $n!=\sum_{k=0}^n \binom{n}{k}(n-k+1)^n(-1)^k$?

I was going through a Number Theory book the other day and found this question. It asked for the proof of the following equation: $$n!=\sum_{k=0}^n \binom{n}{k}(n-k+1)^n(-1)^k$$ I tried hard but ...
2
votes
2answers
57 views

Does this series converge conditionally $\sum_{n=1}^{\infty}\frac{(-1)^n}{n^{\frac{1}{10}}}$

$\sum_{n=1}^{\infty}\frac{(-1)^n}{n^{\frac{1}{10}}}$ According to my understanding, if $\sum\left|a_n\right|$ diverges but $\sum a_n$ converges, then the series is conditionally convergent. For $\...
1
vote
1answer
40 views

Closed form of the summation-$\displaystyle\sum_{r=1}^{n}\frac{r^24^r}{(r+1)(r+2)}$

I have got the following summation-$$\displaystyle\sum_{r=1}^{n}\frac{r^24^r}{(r+1)(r+2)}$$. I have to find the closed form or the general form to find the sum of this series.I know upto summation of ...
0
votes
2answers
76 views

Is there a formula for the binomial expansion of $(a-b)^n$?

Like there is a formula for the binomial expansion of $(a+b)^n$ that can be neatly and compactly be written as a summation, does there exist an equivalent formula for $(a-b)^n$ ?
2
votes
1answer
15 views

Lest number of terms for a partial sum to estimate sum within error

So here is my question. I used the Alternating series to prove that the sum does converge absolutely, so it converges in general. I then tried to say that $\frac{1}{n^4}<10^{-8}$ and solve for n ...
-7
votes
0answers
21 views

The sequence of nubers a1,a2,…an is defined as [closed]

The sequence of nubers a1,a2,...an is defined as an=(1/n+1)-(1/n=2) for each integer n>=1.what is the sum of he first 15 term of this sequence? a. 1/272 b. 1/6 c. 7/16 d. 1/2 e. 15/34 And explain ...
1
vote
2answers
27 views

General Term of a given series , Where $\sum^{n}_{r=1}U_r=\frac{3n}{2n+1}$

General term $U_r $ of a given series , Where $$\sum^{n}_{r=1}U_r=\frac{3n}{2n+1}$$ I can evaluate that $$ U_1=\frac{3}{3}$$ $$ U_2=\frac{1}{5}$$ $$ U_3=\frac{3}{35}$$ $$ U_4=\frac{1}{21}$$ $$ ...
2
votes
1answer
33 views

How to find that a number is a sum of multiple of different numbers?

Suppose a product comes in packs of 3, and 5, and a customer demands 8 quantities of that ...
1
vote
1answer
40 views

Solving series involving Poisson and Binomial

I'm trying to solve $$ \sum_{b=0}^\infty Poisson(b, \lambda)\sum_{x=0}^b binomial(x, b, p)\left(\frac{x}{x+1}\right)^x \\ = \sum_{b=0}^\infty \frac{e^{-\lambda} \lambda^b}{b!}\sum_{x=0}^b {b \choose ...
2
votes
2answers
51 views

Why does $\sum_{i=0}^{n-1} \frac{1}{n-i} = \sum_{i=1}^{n }\frac{1}{i}$?

From CLRS Problem 4.3, part 5 . Why does the following holds? $$\sum_{i=0}^{n-1} \frac{1}{n-i} = \sum_{i=1}^{n }\frac{1}{i}. $$
3
votes
1answer
51 views

How to determine sum of an alternating power series and to prove that sum is positive

I am working on a problem involving an alternating power series as follows: $$\sum_{i=0}^{a-2} (-1)^{a+b-i-2}(a+b-i-1)x^{a+b-i-2}$$ $a$ and $b$ is constant with $0<x<1$ I would like to ...
1
vote
1answer
22 views

Help with simplifying summation

Let $x \in \mathbb{R}^n$ be a vector of non-negative numbers $x = (x_1, \ldots, x_n)$ with mean $\bar{x}$. I would like to prove that $$\frac{\sum_{i = 1}^n \sum_{j = 1}^n \left| x_i - x_j \right|}{2 ...
4
votes
7answers
120 views

A closed form for $1^{2}-2^{2}+3^{2}-4^{2}+ \cdots + (-1)^{n-1}n^{2}$

Please look at this expression: $$1^{2}-2^{2}+3^{2}-4^{2} + \cdots + (-1)^{n-1} n^{2}$$ I found this expression in a math book. It asks us to find a general formula for calculate it with $n$. ...
0
votes
0answers
22 views

Aproximation for the variance (sum)

Given that we know The mean of a population $\tilde W(t) = \sum_{i=1}^{n}f_{i}(t)*W_{i}$ The variance of the population in the previous step $Var(0) = \sum_{i=1}^{n}f_{i}(0)*(W_{i}-\tilde W(0) )$ ...
0
votes
1answer
35 views

Summation Notation (Discrete Mathematics) [closed]

I am currently studying sequence which I think will lead up to my next topic induction. My question is if $$\sum_{k=0}^n \frac{k+1}{n+k}= \frac{1}{n}+\frac{2}{n+1}+\frac{3}{n+2}+\cdots+\frac{n+1}{2n}$...
0
votes
1answer
27 views

Unique integer solutions to $\sum\limits_{i=1}^n a_i = A$ when $l \leq a \leq u$ and $a,A,l,u \in \mathbb{N}$

I'm trying to find a analytical way for finding the total amount of unique solutions to equation: $$\sum\limits_{i=1}^n a_i = A, \text{when } l \leq a \leq u,$$ where $a,A,l,u \in \mathbb{N}$. For ...
2
votes
0answers
41 views

Multiplication of polynomials of the same degree

Consider polynomials of the form \begin{equation} p(x)=x^{n-2r}\sum_{i=0}^ra_ix^{2i}, \end{equation} where \begin{align} r&=n/2, \quad n \quad \text{even},\\ r&=(n-1)/2, \quad n \quad \text{...
0
votes
1answer
30 views

Summations of series double et produit double

I need help with these summations if you can provide me with detailed answers it would be great: $S_{n}=\sum_{k=1}^{n}\left(1+\frac{1}{k^2}+\frac{1}{(1+k)^2}\right)^{1/2}$ $S_{n}=\sum_{k=1}^{n} (k-1)...
0
votes
1answer
35 views

An error in the simplification of a sum

I try to simplify the sum $\sum_{j=0}^{i-2}j3^j$. My method leads to an error. I proceed by evaluating $3(\sum_{j=0}^{i-2}3^j)'$ The derivative $(\sum_{j=0}^{i-2}3^j)'$ is equal to $\left(\frac{3^{i-...
1
vote
1answer
48 views

Prove $\sum_{k=1}^{n} 2^n \text{ mod }k > 2n$ where $n > 1000$

This problem is taken from a Russian textbook of past Olympiads. Its statement looks like this : Given a natural number $n > 1000$ prove that $\sum_{k=1}^{n} 2^n \text{ mod }k > 2n$. ...
0
votes
1answer
38 views

Is it possible to calculate the Average of Products from the Product of Averages?

I have two sets of data - one $X$ measuring the unavailability at a site, the other $Y$ measuring the number of antennas at each site. I want to calculate the overall average unavailability as the ...
1
vote
3answers
50 views

$ z^n = a_n + b_ni $ Show that $ b_{n+2} - 2b_{n+1} + 5b_n = 0 $ (complex numbers)

$$ z = 1+2i \ (complex \ number) \\ z^n = a_n + b_ni \ \ \ (a_n, b_n \in \mathbb{Z}, n \in \mathbb{N}^*) $$ Prove that $ b_{n+2} - 2b_{n+1} + 5b_n = 0$ How can I solve this? Thank you! EDIT: Or ...
0
votes
2answers
68 views

A **proof** for $\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2}$ [duplicate]

I need a proof for the inequality: $\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2}$ for all natural numbers $t \geq 2$. For $t=2$ both sides are equal. Can someone find a proof for all $t$? maybe ...
2
votes
3answers
108 views

A proof for the inequality $\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2} $ for all $t \geq 2$

I'm struggling with proving the following inequality: $$\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2}$$ for all $t \geq 2$. I think it is monotonic non-increasing in $t$, which would suffice. ...
0
votes
1answer
19 views

Single summation with two variables

Disclaimer: This is for homework, but I'm just stuck on this one small part of a larger problem. I'm having trouble figuring out how to get the following summation in closed form. $$\sum_{j=1}^i 4ij$...
1
vote
1answer
42 views

Closed form solution of a hypergeometric sum.

The binomial theorem is one of the best known hyper-geometric sums for whom a closed form expression exists. The natural question is whether generalizations exist . In particular I would like to know ...
-4
votes
1answer
86 views

Definite Integral using its McLaurin Series.

I'm trying to solve the next integral, using its series. However, I got stuck in a very dumb way nearly at the end. The infamous: $$\int_{0}^{1} \frac{\text{cosh}(x)-1}{x}dx$$ First, the series of $\...
0
votes
2answers
101 views

Find the integer part of the sum $S=\sum_{k=1}^{80} \frac{1}{\sqrt k} $

Let $$S=\sum_{k=1}^{80} \frac{1}{\sqrt k}.$$Then I would like to obtain $\lfloor S \rfloor$, the integer part of $S$. I am not able to think how to start question .
6
votes
3answers
121 views

Compute $\sum\limits_{k=0}^{100}\frac{1}{(100-k)!(100+k)!}$

$$\sum_{k=0}^{100}\frac{1}{(100-k)!(100+k)!}$$ My work $$\sum_{k=0}^{100}\frac{2n!}{(2n!)(n-k)!(n+k)!}$$ $$\sum_{k=0}^{100}\frac{^{2n}C_{n-k}}{(2n!)}$$ $$\sum_{k=0}^{100}\frac{^{2n}C_{n+k}}{(2n!)}$$...
2
votes
0answers
31 views

Issues regarding my take on proving $E(X) = \lambda$, where $X\sim Poisson(\lambda)$

My proof: Let $X\sim \mathrm{Poisson}(\lambda)$. Then $$f_{\Tiny{X}}(x) = \frac{\lambda^x}{x!} e^{-\lambda}.$$ Thus, $E(X) = \sum_{x=0}^{\infty} x f_{\Tiny{X}}(x) = \sum_{x=0}^{\infty} x \frac{\lambda^...
27
votes
2answers
808 views

Proof or derivation of this identity $\lim_{n\to \infty}{\frac1{2^n}\sum_{k=0}^n\binom{n}{k}\frac{an+bk}{cn+dk}}\;\stackrel?=\;\frac{2a+b}{2c+d}$?

I just came up with the following identity while solving some combinatorial problem but not sure if it's correct. I've done some numerical computations and they coincide. $$\lim_{n\to \infty}{\frac{1}{...
0
votes
0answers
28 views

Show $\sum_{k=0}^n b_r(n,k) = (r-1)!\frac{x^{\bar{n}}}{(x+1)^{\bar{r-1}}}$ [duplicate]

Let's define $b_r(n,k)$ as $n$-permutations with $k$ cycles where numbers $1\dots r$ belong to one cycle. I tried to first define closed form for $b_r(n,k)$. My idea: We need to put $1 \dots r$ into ...
1
vote
1answer
52 views

How to get the last line?

I'm supposed to find the following equality: $\sum_{r=1}^{s-1}2^{2r-1} -\sum_{r=1}^{s-1}2^{r-1} +(n-2^{s-1}+1)(2^s-1)$ $=\frac{2}{3}(4^{s-1}-1)-(2^{s-1}-1)+(2^s-1)n-2^{2s-1}+3*2^{s-1}-1$ I ...
8
votes
2answers
70 views

Sum to closed form

I need to evaluate the following summation: $$ \sum_{n\in\mathbb{Z}} \frac{-1}{i(2n+1)\pi -\mu} $$ where $n$ is summed over all the integers from $-\infty$ to $\infty$ including 0. Putting this into ...
-6
votes
3answers
91 views

Is $\sum_{n=0}^\infty$ a misleading notation? [closed]

Why do we use the notation $\sum_{n=0}^\infty a_n$, for what is not defined as a sum, but is in fact a limit of a totally different expression? I understand what it means to sum a finite number of ...
1
vote
0answers
42 views

Does the following series converge to what I think it does?

I was wondering if the following series converges: $$\sum_{k=0}^\infty\frac{\Gamma(n+2)}{k!\Gamma(n+2-k)}B_kx^{-k}\tag1$$ And if so, does it converge to $$\frac{n+1}{x^{n+1}}\sum_{k=1}^xk^n\tag2$$ ...
4
votes
2answers
81 views

Find $\sum_{m=0}^n\ (-1)^m m^n {n \choose m}$

I'm going to university in October and thought I'd have a go at a few questions from one of their past papers. I have completed the majority of this question but I'm stuck on the very last part. In ...
1
vote
0answers
40 views

Interchanging sum and integral

If $\mathbf {i)} f_n\in C^1(\overline U)$ ($U$ is convex), $\mathbf {ii)} \sum\limits_{n\ge1}df_n$ converges unifromly on $U$ and $\mathbf {iii)} \sum\limits_{n\ge1}f_n$ converges in an arbitrary ...
4
votes
2answers
142 views

Evaluate $\lim_{n \to \infty }\left(\frac{1}{{n\sqrt{{n^2} + 1}}}+\frac{2}{{n\sqrt{{n^2}+4}}}+\cdots+\frac{n}{{n\sqrt{{n^2}+{n^2}}}}\right)$

Evaluate $$\mathop {\lim }\limits_{n \to \infty } \left(\frac{1}{{n\sqrt {{n^2} + 1} }} + \frac{2}{{n\sqrt {{n^2} + 4} }} + \frac{3}{{n\sqrt {{n^2} + 9} }} + \cdots + \frac{n}{{n\sqrt {{n^2} + {n^2}}...
2
votes
1answer
54 views

Distribution of distances between points with complete spatial randomness

I'm trying to compute the probability of the distances between points on a 2D domain that have complete spatial randomness (CSR). From this wikipedia page on CSR, the probability of locating the $N$...
2
votes
2answers
67 views

Prove by induction that $\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$

As the title says I need to prove the following by induction: $$\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$$ When trying to prove that P(n+1) is true if P(n) is, then I ...
5
votes
3answers
88 views

Factorial Proof by Induction Question? [duplicate]

$\text{Use the PMI to prove the following for all natural numbers n.}$ $ \frac{1}{2!} + \frac{2}{3!} + \cdot \cdot \cdot + \frac{n}{(n+1)!} = 1 - \frac{1}{(n+1)!} $ So for this question I get ...
35
votes
9answers
2k views

Is difference of two consecutive sums of consecutive integers (of the same length) always square?

I am an amateur who has been pondering the following question. If there is a name for this or more information about anyone who has postulated this before, I would be interested about reading up on it....