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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2answers
77 views

How can this summation be proven to be equal?

I have a problem understanding how the following two sides of the Equation can be proven to be Equal to eachother. $$-\frac{1}{h}\sum_{j=0}^{n} [(u_{j+1}-u_j)-(u_j-u_{j-1})]v_j= ...
3
votes
3answers
92 views

Sum of $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{9}+\frac{1}{12}+\cdots$ [duplicate]

My problem is to find the sum of the series $$ S = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{9}+\frac{1}{12}+\cdots $$ where the terms are the reciprocals of the positive ...
0
votes
0answers
65 views

Continuous functions inner product definition

If we have two functions F1(x) & F2(x) defined on [x1,x2] then their inner product is $\int_{x_1}^{x_2} F_1(x)F_2(x)dx$ The origin of the definition is that F1(x) can be represented as a vector ...
1
vote
0answers
27 views

Check my answer - radius and domain of convergence

We want to determine the radius and domain of convergence of the sum $\sum_{n=1}^{\infty}(\frac{2n-1}{3n+2})^n(x+2)^n$ I just want to verify my solution and method. We know that the sum definitely ...
1
vote
1answer
41 views

Find a limit of a function $f$ as $n\to \infty$

$f(n)=\dfrac{1^p+2^p+...+n^p}{n^{p+1}}$. I tried to use Squeeze theorem somehow: $$\dfrac{1^p+1^p+...+1^p}{n^{p+1}}\leq f(x) \leq \dfrac{n^p+n^p+...+n^p}{n^{p+1}}$$ As you can predict I get nowhere. I ...
1
vote
2answers
54 views

Seeking combinatorial proof for $F_{n+1} -1=\sum\limits_{k=0}^{n-1} F_k$

In order to give a combinatorial proof for this equation, we need to find what these two count for. But I don't know what they count for and how I can pivot the RHS to show that it actually counts ...
2
votes
2answers
158 views

Formula for $S_n$

I need help deriving a formula for the nth partial sum of the following infinite series: $$\sum_{n=1}^\infty\,\left[\ln\left(\sqrt{n\,+\,1}\right)\ -\ \ln\left(\sqrt{n}\right)\right]$$ I thought ...
0
votes
1answer
24 views

Approximating a binomial sum over a simplex

For partial binomial sums such as $\sum_{k\le\Delta} \binom{n}{k}$ we don't tend to have closed forms. However we still know asymptotic expansions that are easy to work with. Can we do something ...
4
votes
2answers
44 views

Calculating a rate with a single summation

I have a $n$ events, each with some value $x$ and duration $t$. I can calculate the global rate as follows: $$\text{rate} = \frac{\sum_{i=1}^nx_i}{\sum_{i=1}^nt_i}$$ Is it possible (or probably ...
0
votes
1answer
25 views

Compute convolution

Let $a>0$ and $a_1>0$ be real numbers. Using the convolution theorem I have shown a following identity: \begin{equation} \sum\limits_{l=1}^n \binom{a_1}{n-l} \frac{1}{l} \binom{a}{l} = -\gamma ...
1
vote
1answer
56 views

How many times can a positive integer be written as the sum of $n$ numbers or less, where each addend is between 1 and 9?

I am asking to know how many numbers there are with $n$ digits such that the sum of their digits is $d$. Zero is excluded as a possible digit. I need to prove a problem in which I show that there ...
1
vote
1answer
29 views

Reindexing a Triple Sum with two parameters

I have the following triple summation: $$\sum_{n=0}^{r-3}\sum_{k=0}^{n}\sum_{j=0}^{k}a_j b_{k+s-j}c_{n+t-k}x^{n+s+t}$$ I would like to reindex to have $n$ start at $s+t$ which will make my $x$ power ...
0
votes
1answer
30 views

Reindexing a Triple Sum

Suppose i have the following integral: $$\sum_{n=0}^{4}\sum_{k=0}^{n}\sum_{j=0}^{k}a_{n+2-k}b_{j}c_{k-j}x^{n+2}$$ I want to reindex to make the power of $x$ simply $x^n$, so I have to shift by 2. ...
-2
votes
1answer
63 views

How to derive this summation formula? [closed]

$$\sum_{i=0}^{n-1} ia^i = \frac{a-na^n+(n-1)a^{n+1}}{(1-a)^2}$$ What is the thought process behind obtaining this formula?
1
vote
1answer
32 views

Are these equivalent summations?

I just want to ask if $$\sum_{i=1}^n \sum_{j=1}^i |a_i \bar b_j-a_j\bar b_i|^2 = \sum_{i=1}^n \sum_{j=1}^n (a_i \bar b_j-a_j\bar b_i)(\bar a_ib_j + \bar a_j b_i)$$ is true and possibly an ...
1
vote
3answers
53 views

How to prove $\left(\sum_{i=1}^n x_i\right)^2 \le n\sum_{i=1}^n x_i^2$?

For now I have the following \begin{align*} \left(\sum_{i=1}^n x_i\right)^2 & \le n\sum_{i=1}^n x_i^2 \\ \sum_{i=1}^n x_i^2 + 2\sum_{i<j}x_ix_j & \le n\sum_{i=1}^n x_i^2 \\ ...
0
votes
1answer
23 views

Given any subset of [1, 2, …, n], can its sum mod (n) + 1 be changed to any value between 1 and n?

Given any subset of $[1, 2, ..., n]$, can its sum $\bmod n + 1$ be changed to any value between $1$ and $n$ by adding a value from the original set or removing a value from the subset? For example: ...
1
vote
1answer
20 views

Triple Sum construction

I have the following: $$a_3b_0c_0+a_3b_1c_0+a_3b_0c_1+a_4b_0c_0+a_3b_2c_0+a_3b_1c_1+a_3b_0c_2+a_4b_1c_0+a_4b_0c_1+a_5b_0c_0$$ I can rearrange then to get $$a_3b_0c_0+$$ ...
5
votes
4answers
85 views

Limit of $(\sum_{k=0}^{n}k^4)/n^5$

So i was trying to find this limit: $$\lim_{n\to\infty}\frac{ \sum_{k=0}^{n}k^4}{n^5}$$ which at first made me think it's zero but soon i realized that it's probably not. I tried expanding that but ...
2
votes
3answers
58 views

How can I compute $\sum\limits_{k = 1}^n \binom{n - 1}{k - 1}$?

I know what $n \choose k$ equals, but I don't see how that would help me solve the sum of $n - 1 \choose k - 1$ from $k = 1$ to $n$. Is there any special trick I should know?
0
votes
1answer
23 views

What is the formula for sum of $(n-r+1)C_r$ for a given $n$?

I wish to calculate the sum of $$(n-r+1)C_r$$ for a given $n$. Example: For $n=6$, the sum equals $6C_1 + 5C_2 + 4C_3$. I have a very large $n$ $(n\ge 10^6)$. Help please.
3
votes
1answer
337 views

How to deal with this double summation?

I'm stuck with the proof of this result: $$2^n = \sum_{t=-\frac{n-1}{2}}^{\frac{n-1}{2}} \binom{n+1}{\frac{n+1}{2} + t} \sum_{k=\vert t \vert}^{\frac{n-1}{2}} \binom{\frac{n-1}{2}+k}{k} ...
2
votes
2answers
68 views

Sum of $\lfloor k^{1/3} \rfloor$

I am faced with the following sum: $$\sum_{k=0}^m \lfloor k^{1/3} \rfloor$$ Where $m$ is a positive integer. I have determined a formula for the last couple of terms such that $\lfloor n^{1/3} ...
0
votes
1answer
33 views

Given $g(x)$, find $f(x)$, knowing $f(x) = \sum_{a=1}^x g(a)$

Given $g(x)$, find $f(x)$, knowing $f(x) = \sum\limits_{a=1}^x g(a)$ Is there a universal approach of finding $f(x)$, regardless of $g(x)$? For simplicity sake, assuming that $g(x)$ is a polynomial ...
1
vote
3answers
95 views

Show that $ \sum_{k=0}^{\infty} (k-1)/2^k = 0$

I am a computer scientist trying resolve exercises from CLRS. Here is one that I can't make progress on. Show that $\sum\limits_{k=0}^{\infty} (k-1)/2^k = 0 $ What I did so far: $$ ...
1
vote
1answer
44 views

A newbie trying to understand as how this summation was simplified

Can someone tell me as how this summation works? I tried simplifying using geometric progression. But I could not Thank you
0
votes
1answer
33 views

Question involving summations and the Θ-notation of running times

I think I understand the concept of summations and Θ-notations, however, I don't really understand the question below. If I have understood it correctly, I'm supposed to write out the summations ...
0
votes
2answers
26 views

Prove simplification of summation

I have the following equation: $$ \sum_{i=1}^n 2-\frac{2i}{n} $$ When running it with different values of $n$, the result seems to always be $ n - 1 $ so that $$ \sum_{i=1}^n 2-\frac{2i}{n} = n - 1 ...
11
votes
1answer
386 views

Challenging identity regarding Bell polynomials

Note: [2015-03-08] A proof of the identity below was aimed to close the gap of a rather extensive elaboration of this answer of mine. The identity (1) below is part of a more complex one, which is ...
4
votes
0answers
67 views

If $f$ is integrable, then $\sum\limits_{n\ge 1}\frac{1}{\sqrt n}\vert f(x-\sqrt n)\vert$ is almost everywhere finite

I would like to show that $$\sum_{n\ge0}\left\vert \frac{1}{\sqrt n} f \left(x-\sqrt n \right)\right\vert \tag{$*$}$$ converges for almost every (a.e.) $x$. The only technique I have is based on the ...
2
votes
0answers
34 views

Simpler expression for binomial sum

Is there any closed expression for the following sum: $$\sum_{i=0}^{l-k} \binom{n-l}{i} \binom{l-k}{i} \binom{l-i}{k}$$ where $ k<l < n/2$?
0
votes
1answer
60 views

How to calculate $\frac{1}{n}\sum_{i=1}^{n}\frac{i}{i+1}$

I have been working with some probabilities and I am not sure if I'm correct but one of my probabilities is reduced to this formula, however, I don't recall a method to solve it: ...
2
votes
0answers
46 views

Trouble writing Double Summation

I have the following: ...
3
votes
1answer
31 views

Is $\sum_{n=1}^\infty \frac{\left(-1\right)^{n+1}\log\left(n\right)}{n}$ divergent?

Is \begin{align}\sum_{n=1}^\infty \frac{\left(-1\right)^{n+1}\log\left(n\right)}{n}\tag{1}\end{align} divergent? I think it is because by comparison \begin{align} \sum_{n=1}^\infty ...
1
vote
2answers
23 views

Does $\sum_{n=1}^{\infty}\left(-1\right)^{n+1}\frac{3\sqrt{n+1}}{\sqrt{n}+1}$ Diverge or Converge?

Does the following summation converge or diverge? \begin{align} \sum_{n=1}^{\infty}\left(-1\right)^{n+1}\frac{3\sqrt{n+1}}{\sqrt{n}+1}.\tag{1} \end{align} I don't know where to begin. I think I ...
2
votes
0answers
36 views

Using Substitution for Convolution

Suppose I have the following product: $$\sum_{k=0}^{a}\alpha_kx^k\sum_{k=0}^{b}\beta_kx^k\sum_{k=0}^{c}\gamma_kx^k$$ Note that the bounds are finite and not equal; $a\neq b \neq c$. I'm looking to ...
0
votes
1answer
13 views

Linking summations with their correct function(s)

Guys can you please guide me step by step on how to link given functions with the functions to choose from. So for example a function $g(n)\in \Theta n^2$ and if there is no match then you say there ...
0
votes
1answer
81 views

Summation of factorial series [duplicate]

I want to calculate the sum of this series: $$S = 1\cdot1! + 2\cdot2! + 3\cdot3! + 4\cdot4! +\dots+ n\cdot n!$$ Is their any formula for finding this sum?
0
votes
1answer
23 views

Help with testing for convergence

Let $(a_k)$ denote a real sequence. Use the indicted test to show if $\sum a_k$ converges. $\frac{(2k+1)(3k-1)}{(k+1)(k+2)^2}$ using Comparison Test of the limit form $(-1)^{k-1}\sqrt{k+1}-\sqrt{k}$ ...
0
votes
1answer
28 views

Cumulative binomial distribution sum manipulation

I have a binomial distribution, with Random Variable Y and n trials. r is an integer. How can I show that P(Y ≥ r) = P(X ≤ n − r), such that Y is a random variable with probability of success p, and X ...
1
vote
0answers
25 views

Did i multiply the sums correctly?

This is an extention to this question except i am unsure of whether i have done it correctly: $$ y'' = -y'(f(x) - r(x) y') $$ $f(x) = \sum_{n=0}^\infty s_n x^n$, $y = \sum_{n=0}^\infty a_n x^n$, and ...
0
votes
2answers
116 views

Sum of 16 unsigned integers, possible combinations.

I have two arrays with 16 unsigned integers. I compute the sum of the first array = x and the sum of the second array = y. What is the chance they will be the same? Also, how many combinations out of ...
5
votes
2answers
65 views

Induction Proof [duplicate]

Prove that: $1^3+2^3+...+n^3=\frac{n^2(n+1)^2}{4}$ for $n \in N$ So I am thinking that I need to do a proof by mathematical induction. Here's my attempt: Let S(n) be the statement ...
4
votes
2answers
62 views

How can we find the closed form of this?

$$\lim_{n\to\infty}\frac1n\sum_{k=1}^n\left(\left\lfloor\frac{2n}k\right\rfloor-2\left\lfloor\frac nk\right\rfloor\right)$$ I think it is equal to $2(\frac13 - \frac14 + \frac15 - \frac16 + \cdots)$, ...
0
votes
0answers
38 views

Why is my approximation of this alternating series incorrect?

I've been working on some calc problems and I'm stuck on the second part of a problem consisting of estimating the value of a series with a given error. I tried calculating it by hand.. wolfram, ...
1
vote
1answer
118 views

The sum of $1+1+1+1+…$

My teacher recently showed me a rather weird result and I would like to know if he was just tricking me or if he was serious. He showed me that $g=1-1+1-1+1-...=\frac{1}{2}$ Then he said that ...
3
votes
4answers
76 views

Summation Notation: $i<j$

Is this true? $$ \sum\limits_{i<j} x_ix_j = \sum\limits_{j=1}^n \sum\limits_{i=1}^j x_ix_j$$ $i, j = 1,\ldots,n$ And on the left hand side, how can you tell when it stops?
2
votes
1answer
40 views

Finding a paper/reference where a solution also exists

I have already calculated the following sum: $$\sum_{k=0}^\infty{B_2(k)}=\frac{\pi^2}{3}-2\approx1.2898681336964$$ where $B_2(k)$ is the 2nd hypergeometric Bernoulli number of order $k$. These ...
0
votes
3answers
42 views

How to simplify this summation?

I am working through an example in my book, and I can't seem to figure out how they go from one step to another in a particular case. Here is the example: (note: lg here is just the base-2 logarithm, ...
1
vote
2answers
59 views

Legendre polynomial to show identity, can't spot mistake

Using Legendre polynomial generating function \begin{equation} \sum_{n=0}^\infty P_n (x) t^n = \frac{1}{\sqrt{(1-2xt+t^2)}} \end{equation} Or $$ P_n(x)=\frac{1}{2^n n!} \frac{d^n}{dx^n} [(x^2-1)^n] ...