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

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5
votes
4answers
82 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
55 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
21 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
306 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} ...
1
vote
1answer
43 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
32 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
85 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
25 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
23 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
368 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
65 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
30 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
58 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
45 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
35 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
0answers
11 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
60 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
22 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
25 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
93 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
62 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
61 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
34 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
110 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
73 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
39 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
45 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] ...
2
votes
1answer
25 views

reverse of a summation formula

This is the formula: $\sum_{i=1}^n x^i = a$ For example: $\sum_{i=1}^4 3^i = 120.$ If we have x and a, is it possible to ...
1
vote
1answer
48 views

Proving the Fibonacci identity $\sum_{i=1}^n f_i^2=f_nf_{n+1}$ by induction [duplicate]

I am having troubles with a proof question. Prove that for any $n\ge1$, $\sum_{i=1}^n f_i^2=f_nf_{n+1}$, where $f_n$ is the $n$'th Fibonacci number. I have the base case and the induction ...
0
votes
2answers
43 views

Legendre Polynomial manipulations

Given Legendre polynomial generating function \begin{equation} \sum_{n=0}^\infty P_n (x) t^n = \frac{1}{\sqrt{(1-2xt+t^2)}} \end{equation} Show that $$ P_n (1)=1 $$ and $$ P_n (-1)=(-1)^n $$ ...
0
votes
3answers
93 views

Deducing a $\cos (kx)$ summation from the $e^{ikx}$ summation [duplicate]

I'm trying to solve So far I've done the first part, evaluating the summation ; where a is just n. I'm not sure where to go from here or what it even means deduce the second summation. I ...
2
votes
1answer
51 views

Show that there is exactly one positive integer n for which $\sum_{r=1}^n r^3+\sum_{r=1}^n r = 8 \sum_{r=1}^n r^2$

Can someone show me the working for this? Thanks AQA A Level Mathematics Further Pure 1 January 2010 Question 8(b) http://filestore.aqa.org.uk/subjects/AQA-MFP1-W-QP-JAN10.PDF And if anyone could ...
0
votes
0answers
33 views

Why is the following equality true?

I am trying to understand a proof - why does this hold? $$\sum\limits_{i}\left(\sum\limits_{0 \le j \le ...
0
votes
1answer
44 views

Summation of harmonic series. [closed]

I'm trying to figure out how to answer this linear algebra question and can't figure it out. Can someone please explain it to me? Thanks a bunch! Here's the questions:
1
vote
0answers
22 views

Did i generalize this series solution correctly?

$$ f(x) = \frac{y''}{p(x)y'} + r(x) y' $$ if all functions are expressed in their power series form, then: $$ y = \sum_{n=0}^\infty a_nx^n $$ $$ p(x) = \sum_{n=0}^\infty p_n x^n $$ $$ r(x) = ...
1
vote
1answer
38 views

How to solve this triple summation problem?

For a computer science class we were asked to analyze the run time of an algorithm. The answer was posted. I am not sure the proof is correct. I believe the answer should be kc(n^2) (where k is a ...
0
votes
0answers
29 views

Question about multiplying summations with another summation inside

I have the following: $$ y = \sum_{n=0}^\infty [x^n \sum_{k=0}^\infty (k+1)a_{k+1} P_{n-k}] \sum_{n=0}^\infty x^n[s_n - \sum_{k=0}^n a_{k+1}(k+1)R_{n-k}] $$ I can easily multiply $$ ...
3
votes
1answer
166 views

Hard binomial sum [closed]

How to prove this relation? $$\sum_{i=0}^{n}\frac{2^{-2i}\binom{2i}{i}}{n+i+2}=\frac{2^{4n+2}-\binom{2n+1}{n}^2}{(2n+3)2^{2n+1}\binom{2n+1}{n}}$$ That seems difficult!
0
votes
1answer
35 views

Independent summantion trick?

We know that: $$\sum_{k}\left(f(k)\sum_lg(l)\right)=\left(\sum_kf(k)\right)\left(\sum_lg(l)\right)$$ Since both counters are independent: But what if we have: ...
0
votes
0answers
15 views

Bound from distinct integer summation

We want to find $r$ positive integers $\{a_i\}_{i=1}^r$ such that of atmost $(s+1)^r$ values obtained from $$\sum_{i=1}s_ia_i$$ where $s_i\in\{0,\dots,s-1,s\}$, we insist on some combination of ...
2
votes
1answer
89 views

Sum over subsets of $\{1,2,\ldots,n\}$ of terms involving a product over that subset

I'm attempting to perform a sum, using products, using all possible combinations, in a function. How would I go about doing this? (I really need to find something that works.) For example, say I ...
5
votes
5answers
59 views

Demonstration of sum of powers of $2$ [duplicate]

Theorem : For every natural number $p$: $$\sum^p_{i=0} 2^i = 2^{p+1}-1$$ I trieed to demonstrate the theorem using induction Demonstration : $1)$ If we have $p=0$ then we get $2^0=2^{0+1}-1$ that is ...
3
votes
0answers
54 views

$\lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$

I am just having fun with this question: Is this true that $\displaystyle \lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$? I thought to change this to an integral, namely ...
0
votes
0answers
20 views

Discrete Mathematics; Counting, Summations [duplicate]

Let n ≥ 1 be an integer. Prove that: $$ \sum\limits_{i=1}^n i(\frac{n}{i}) = n \bullet 2^{n-1} $$ I am not sure how to prove this, I think I need to use the derivative of $$(1 + x)^ n$$ any help ...
2
votes
1answer
33 views

Limit to zero of the $p$-norm

I have the $p$-norm defined as $$\|x\|_p=\left(\sum_{k=1}^n|x_k|^p\right)^\frac{1}{p}.$$ I am trying to find the limit as $p\to0^+$ of $\|x\|_p$. I've seen it defined as $\{x_k:x_k\neq0\}$. Why is ...