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

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0
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5answers
61 views

Sum of $n$ terms of series

What is the sum of $n$ terms of the series $${1\over(4\times 9)} + {1\over(9\times 14)} + {1\over(14\times 19)} + {1\over(19 \times 24)} + ... ?$$ The answer is $n\over 4(5n+4)$, but I can't figure ...
0
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0answers
21 views

Derivation of least squares for a line $y=a+bx$

I was trying to obtain the formula for the least squares regression for a line: I'm not able to compute the formula that gives the errors on the two parameter. For the "true value" I obtained for ...
0
votes
0answers
16 views

Summation closed form for $k^{n^p}$

Is there any closed form expression for $$\large\sum_{n = 0}^{\infty} k^{n^p} / \sum_{n=0}^{\infty} k^{-n^p}$$ I know that for $p=1$, this converges to $-k$. Is there any other known solution for ...
3
votes
1answer
29 views

Sum of the first $n$ terms of polynome

There are several related questions here, but none seem to have helped me so far. There is an exercise in my book that goes like this: Determine the sum of the first $n$ elements of $$x^3 + x^5 + x^7 ...
2
votes
2answers
56 views

Find sum with binomial coefficients and powers of 2

Find this sum for positive $n$ and $m$: $$S(n, m) = \sum_{i=0}^n \frac{1}{2^{m+i+1}}\binom{m+i}{i} + \sum_{i=0}^m \frac{1}{2^{n+i+1}}\binom{n+i}{i}.$$ Obviosly, $S(n,m)=S(m,n)$. Therefore I've tried ...
-2
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1answer
31 views

$n k^n$ summation question [closed]

How does one prove that Can this be extended to higher powers such as: Thanks!
1
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1answer
49 views

The logic behind rearranging summations - generating functions

I'm learning about generating functions, and one part that I am struggling with is the logic behind rearranging summations (particularly double summations). I'll illustrate an example: Using the ...
1
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4answers
261 views

How to simplify this triple summation containing binomial coefficients?

$$ \large\sum_{i=0}^{n} \sum_{j=i}^{n} \sum_{k=j}^{n} \binom{i+m-1}{m-1}\binom{j+m-1}{m-1}\binom{k+m-1}{m-1} $$ How to solve it when this involve more than thousand summation ?
0
votes
1answer
27 views

Ways to further simply recursive relation

I was working on a power series solution in my ODE class and I had found that my $a_n$ seemed to be defined as $$a_n=\frac{a_o}{(n^{2}(n-1)^{2}…(n-(n+2))^{2}}$$ but I am having trouble understanding ...
3
votes
2answers
121 views

Is there a closed form for $\sum_{n=0}^{+\infty} \frac{1}{\sqrt {n!}}$?

This is a curiosity question. Let's consider the following sum: $$S=\sum_{n=0}^{+\infty} \frac{1}{\sqrt{ n !}}$$ The question asked to prove its convergence, which I did using the ratio test. So I ...
4
votes
0answers
118 views

How to solve this multiple summation?

How to solve this summation ? $$\sum_{0\le x_1\le x_2...\le x_n \le n}^{}\binom{k+x_1-1}{x_1}\binom{k+x_2-1}{x_2}...\binom{k+x_n-1}{x_n}$$ where $k$ , $n$ are known. Due to hockey-stick identity , ...
2
votes
1answer
25 views

Finding a bound on double summation involving primes

I am reading a number theory proof of a result in which I am stuck on a bound.Suppose $p_1$ and $p_2$ are primes with the property that each $p_i$ satisfies $e^r \leq P_i <e^{r+1}$ and $P_1 \equiv ...
2
votes
1answer
61 views

Evaluate $\lim\limits_{x\to\ 1}\frac{x+x^2+\cdots+x^n-n}{x-1}$ without L'Hospital's rule [duplicate]

I used substitution $$t=x-1, x=t+1,x\rightarrow1\Rightarrow t\rightarrow 0$$ Now the expression is $$\lim_{t\to\ 0}\frac{t+1+(t+1)^2+\cdots+(t+1)^n-n}{t}$$ Can we use the sum of geometric sequence ...
1
vote
2answers
47 views

Proof by counting two ways

Proof by counting two ways: \begin{equation}\sum_{k_1+k_2+...+k_m=n}{k_1\choose a_1}{k_2\choose a_2}...{k_m\choose a_m}={n+m-1\choose a_1+a_2+...+a_m+m-1}\end{equation} I have a proof by algebra for ...
2
votes
4answers
86 views

Find $x_{n}$ if $x_{1}=a>0$ and $x_{n+1}=\frac{x_{1}+2x_{2}+…+nx_{n}}{n}$

Find $x_{n}$ if $x_{1}=a>0$ and $x_{n+1}=\frac{x_{1}+2x_{2}+...+nx_{n}}{n}$ I have a problem finding sum of $$x_{1}+2x_{2}+...+nx_{n}$$ I don't see the term $x_{2}$ because if $x_{1}=a$ for ...
3
votes
1answer
36 views

“Commutativity” of sums

In general, is $\sum_i \sum_j f(i,j) = \sum_j\sum_i f(i,j)$ ? With $f(i,j)$ I mean some expression that depends on $i$ and $j$. If yes, how could I prove that?
8
votes
2answers
89 views

Rough bound for sum $\binom{3n}{0}+\binom{3n}{1}+\cdots+\binom{3n}{n-1}$

Is it true that $$\frac{\dbinom{3n}{0}+\dbinom{3n}{1}+\cdots+\dbinom{3n}{n-1}}{2^{3n}}<\frac13$$ for all positive integers $n$? I've plotted the first few values of $n$ and noticed that the ...
2
votes
3answers
31 views

$\sum^6_{i=1}(x_i-\bar{x})^2$ as $\sum^6_{i=1}x_i^2 - 6\bar{x}^2$ what rules where applied?

consider the set $X = \{20, 30, 40, 50, 60, 70\}$ and the mean $\bar{x} = 45$ then $\sum^6_{i=1}(x_i-\bar{x})^2 = 1750 = \sum^6_{i=1}x_i^2 - 6\bar{x}^2$. How would I transform the first term by hand ...
1
vote
2answers
47 views

Ratio of two summations

I devised this question based on recent (and not-so-recent) MSE questions on summations. Evaluate ...
0
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1answer
23 views

time complexity of an algorithm

Hi all i'm trying to predict/calculate the time complexity of an algorithm but i'm having some difficulties with the summations the algorithm: ...
2
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0answers
21 views

evaluation summation Erlang distribution

I have to calculate a renewal function: $H(t)$. In which: $$ H(t)=\sum_{n=1}^\infty \left(1-e^{-\lambda t}\sum_{i=0}^{2n-1}\frac{(\lambda t)^i}{i!}\right) $$ I think it can be solved by switching ...
1
vote
2answers
35 views

Which is greater ? Sum of odd power terms or even power terms in the exponential Taylor series?

I came across this question, in a book. Define $f(x) = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{{(2n+1)}!} $ and $ g(x) = \sum_{n=0}^{\infty} \frac{x^{2n}}{{(2n)}!} $, where x is a real number. Then, ...
0
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0answers
42 views

sum of zeta function [duplicate]

how do I solve this question? $$\sum_{k\geq2} ( \zeta(k)-1) $$ I know that $\zeta(2)$ is $\frac{\pi^2}{6}$ and $\zeta(k)$ can be represented as $$\sum_{j\geq1} \frac{1}{j^k}$$ Thanks in advance!
0
votes
1answer
27 views

On a trigonometric summation.

I have a function defined as: $$C_n(w) = \frac{1}{n} \sum_{j = 0}^{n-1}\sum_{k=0}^j (a_k \cos(kw) + b_k \sin(kw)) $$ Now it is stated that: $$C_n (0) = \frac{a_0}{2} + \frac{1}{n} \sum_{r = 2}^n ...
1
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4answers
96 views

Calculate the sum $\sum_{n=3}^{\infty}\frac{4n-3}{n^3-4n}$

$$\sum_{n=3}^{\infty}\frac{4n-3}{n^3-4n}$$ I think it is related to power series, because it is the topic, but I have no idea how to get there. Could you give a hint?
13
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4answers
297 views

How to prove $ \sum_{k=0}^n \frac{(-1)^{n+k}{n+k\choose n-k}}{2k+1}=\frac{-2\cos\left(\frac{2(n-1)\pi}{3}\right)}{2n+1}$

How to prove $$\sum_{k=0}^n \binom{n+k}{n-k}\frac{(-1)^{n+k}}{2k+1}=-\frac{2}{2n+1}\,\cos\left(\frac{2(n-1)\pi}{3}\right)\;\text{?}$$ I have a proof by induction for it, but it isn't simple! I want ...
0
votes
1answer
85 views

About a sum involving factorials.

I would like to know if there is a closed form of $$\sum_{k=0}^{n}\frac{4^{k}}{\left(2k\right)!\left(n-k\right)!^{2}}.$$ Wolfram gives a strange closed form and, i.e., ...
1
vote
2answers
98 views

If a vector v is an eigenvector of both matrices A and B, is V an eigenvector of A+B? [closed]

If so, is there a proof for this? I have been stuck trying to validate the statement and would love some insight.
6
votes
2answers
101 views

Changing-sided dice probability problem.

Suppose you roll a fair $6$-sided dice, and that the number you roll is $m$. If $m=1$, stop. Otherwise, roll an $m$-sided dice. The number you roll is $n$. If $n=1$, stop. Otherwise roll an $n$-sided ...
0
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1answer
20 views

Question about double summation notation.

Just started learning about double integrals literally $10$ minutes ago. I have a fairly good grip on the Riemann integral and so far it seems very similar, but we are just working with volumes ...
0
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1answer
25 views

Combinatorics-Summation doubt in the proof of the expectation of the Hypergeometric distribution.

The proof starts considering this equality: $(d/dx (1+x)^A)(1+x)^B = A(1+x)^{A+B-1}$ Then it keep on changing every $(1+x)^{A or B}$ for its binomial coefficient. That 's what I don't understand. If ...
16
votes
5answers
111 views

Geometrical interpretation of $(\sum_{k=1}^n k)^2=\sum_{k=1}^n k^3$

Using induction it is straight forward to show $$\left(\sum_{k=1}^n k\right)^2=\sum_{k=1}^n k^3.$$ But is there also a geometrical interpretation that "proves" this fact? By just looking at those ...
2
votes
1answer
33 views

Find the probability generating function $G(s)$ of this branching process.

Suppose that $X_n$ is size of the $n$th generation of a branching process started from a single individual, where each individual has a random number of children with probability mass function: ...
6
votes
0answers
100 views

Partial sums of falling factorials

I want to know if there exists some way, approximate or exact, to do a partial sum of falling factorials of the kind: $$\sum_{k=i}^{n}(a+k)_{h}$$ where all are constants. And I'm interested too in ...
1
vote
1answer
31 views

Looking for help in regard to Series solutions with ordinary points (ODE)

I have a question that is in regard to the final answer that one is to get when solving some ODE questions via series. I am having some confusion on what if I am doing is correct/ why it is or is not ...
2
votes
2answers
47 views

How does this manipulation of summations work?

I am reading some mathematics in which is the following algebraic manipulation. $$ \begin{align} \exp(x)\exp(y) & = \left(\sum_{n = 0}^\infty \frac{x^n}{n!}\right) \left(\sum_{m = 0}^\infty ...
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0answers
40 views

Sum of powered numbers

Here is what I have achieved so far: $S(n,p)= 1^p+2^p...+n^p$ By arranging it this way: $\ \ \ \ \ \ \ \ 1^{p-1}$ $\ \ \ \ 2^{p-1} 2^{p-1}$ $3^{p-1} 3^{p-1} 3^{p-1}$ $..............$ I ...
3
votes
3answers
97 views

evaluate the sum $\sum_{n=1}^{\infty}\sum_{k=n}^{\infty}\frac{1}{(n^2+n-1)(k^2+k-1)}$

I'm trying to evaluate this sum $$\sum_{n=1}^{\infty}\sum_{k=n}^{\infty}\frac{1}{(n^2+n-1)(k^2+k-1)}$$ I have no idea how to deal with it. With one sum I can, with partial-fraction decomposition, ...
2
votes
2answers
65 views

Nested sum $\sum_{i<j< \cdots < k} ij \cdots k$

I am wondering if there is any known closed form for the following nested sum? : $$ \sum_{i<j<\cdots <k} ij\cdots k $$ where each $i,j,\cdots,k =1, \cdots, n$ I tried the first one: $$ ...
2
votes
2answers
52 views

Alternating Sum of Cubes [closed]

How is it possible to evaluate: $$\sum_{k=1}^n{((-1)^{n-k}\cdot k^3)}=n^3 - (n-1)^3 + (n - 2)^3 - \cdots \pm 1^3$$ The fact that there is the $\pm$ at the end makes it difficult.
2
votes
1answer
45 views

How do we derive the sum of $3^n$ and $2^n$

I know that $\quad\sum2^n = 2 (2^n-1)$ How can we derive this summation? And also how can we deduce the summation of $3^n$ from this ? I did observe this pattern : $$ \begin{align} n &= 1 ;\ ...
4
votes
2answers
54 views

Multiplication Principle and Inclusion-Exclusion: $2^n = \sum_{i = 0}^n (-1)^i \binom{n}{i} \binom{2n - 2i}{n - 2i}$

I began to compose an unnecessarily complicated answer to this question: If we had 25 people all who have 2 different balls, how would you work out how many combinations there would be if we want ...
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vote
1answer
35 views

Understanding summations with Poisson

I'm currently doing a problem on Poisson processes and I've encountered the situation where I'm not sure why this summation is expanded as follows: And similarly I have tried expanding out the ...
3
votes
1answer
53 views

Exact value of a sum involving harmonic numbers

Could somebody tell me the exact value of this series? $$ \sum_{k=1}^{\infty} (-1)^k\frac{H_k^{(5)}}{k} $$ where $$ H_k^{(n)}=\sum_{i=1}^{k}\frac{1}{i^n} $$ Thanks!
0
votes
1answer
26 views

Factoring constant in summation

Trying to show that adding a constant c to $\sum_{k=0}^\infty a^kx_k$, where a is a constant will just add some constant k to the summation eg. $$\sum_{n=0}^\infty a^n(x_n+c)$$ $$=k + ...
2
votes
4answers
69 views

How can I simplify $1\times 2 + 2 \times 3 + .. + (n-1) \times n$ progression? [closed]

I have a progression that goes like this: $$1\times 2 + 2 \times 3 + .. + (n-1) \times n$$ Is there a way I can simplify it?
1
vote
1answer
28 views

Expanding a term with a sum

We have the following quantity: $$E\left[\left(\sum^n_{j=1} (X(t_j) - X(t_{j-1}))^2-t\right)^2\right]$$ My textbook says this can be expanded in the following way (colors are my touch) ...
1
vote
2answers
81 views

Closed form of sum with binomial

I want to find closed form of the following expression : $$\sum\limits_{k=0}^{n} \binom{n}{k}\frac{(-1)^k}{2k+1}$$ I have no idea how to do it.
0
votes
2answers
34 views

Approximate summation of flooring function

I have this summation: $\sum_{k=0}^x \lfloor{\frac{k}{c}}\rfloor$ Do you have any ideas on any general expressions that can approximate this? P.S I know I can approximate it with a Fourier ...
1
vote
1answer
44 views

Sum of the series with Stirling numbers of the first kind.

Yesterday I worked on one problem in discrete math and in the process of decision I came across this series. Try to do it with generating functions, but there is no success for me. So, what do you ...