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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0answers
10 views

Simplify the following summation involving the Floor function

Let $x,y,n \in \mathbb{Z}$ and $a\in [0,1/3).$ Further assume that $x<0,$ and $y>-2x.$ Is there any significant way to simplify the following: $\left(\sum\limits_{i=\lceil 1/3-(x+a) ...
6
votes
2answers
56 views

Is this summation solvable? $S_n = \sum_{i = 1}^{n}\log_i{(n)}$

Is it possible to solve a summation with a variable base of log? $$ S_n = \sum_{i = 2}^{n}\log_i{(n)} $$ Should I use the derivative of $\log_i{(n)}$?
2
votes
4answers
55 views

How to find answer to the sum of series $\sum_{n=1}^{\infty}\frac{n}{2^n} $

I have put his on wolfram and obtained answer as follows: $\sum_{n=1}^{\infty}\frac{n}{2^n} = 2$ And the series is convergent too because $\lim_{n\to\infty} \frac {n}{2^n} = 0$ However I am ...
0
votes
2answers
34 views

Calculate the following sequence $\sum_{n=0}^{+\infty }\left ( -\dfrac{1}{4\alpha } \right )^{n}\dfrac{ (2n)!}{n!},\; \alpha >0$

Calculate the following sequence $$\sum_{n=0}^{+\infty }\left ( -\dfrac{1}{4\alpha } \right )^{n}\dfrac{ (2n)!}{n!},\; \alpha >0$$
0
votes
1answer
22 views

Find a bound for the summation $\sum_{j=k}^J jc^{k - j - 1} $

The problem: I've hit what might be a dead-end. If it is true, I would like to show that for $c \in (0,1)$ and $1 \leq k \leq J$, the sum $$ \sum_{j=k}^J jc^{1-j-k} = \sum_{n=1}^{J+1-k} (k-n-1) ...
1
vote
1answer
15 views

Dividing summations that have existing properties in each element

If $a_i/c_i > B$ for all $1 \le i \le k$, is it fair to assume that $(a_1 + a_2 + \cdots + a_k)/(c_1 + c_2 + \cdots + c_k) > B$ ? Is there a way to prove this? Thanks!
10
votes
3answers
952 views

Is this already an equation/law that has been found?

So I was messing around with some numbers today and I have found a way to quickly add summations (probably not the first one to discover it but...) this only works when you start at 1 (i.e. ...
2
votes
1answer
27 views

Question on the Prime Number Theorem (the Tchebychev Function)

This has been giving me nothing but a headache: Let the Tchebychev Function, $\psi (x)$ be defined: $$\psi (x) = \sum_{p^m \le x}\log p \space \space \space , \space \space \space p \in \mathbb P$$ ...
8
votes
2answers
118 views

To compute $\tan1-\tan3+\tan5-\cdots+\tan89$, $\tan1+\tan3+\tan5+\cdots+\tan89$

How do we compute : $$i)\ S_1 = \tan1-\tan3+\tan5-\cdots+\tan89$$ and $$ii)\ S_2 = \tan1+\tan3+\tan5+\cdots+\tan89$$ all the angles are in degrees. Thanks
1
vote
1answer
48 views

Expanding a product formally.

Let $a_1,...,a_n$ be real numbers. I don't know how to formally expand the following product $$ \prod_{k=1}^n(1+a_k) $$ I'm guessing something like (edited) $$1+\huge\sum_{k=1}^n \; ...
1
vote
0answers
52 views

Closed form expression for an infinite sum of a product of matrices

Given a square matrix $A$ with certain properties ($A$ is diagonalizable, $\rho(A) = 1$ and each column of $A$ adds up to unity) and two square matrices $M$ and $N$ of rank $1$ (both matrices are ...
2
votes
2answers
53 views

find sum of first 2002 terms

if $\left \{ a_n \right \}$ is sequence of Real Numbers for $n \ge 1$ such that \begin{equation} a_{n+2}=a_{n+1}-a_n \tag{1} \end{equation} \begin{equation} \sum_{n=1}^{999} a_n=1003 \tag{2} ...
0
votes
0answers
17 views

Does this recursion/sequence of iterated infinite sums converge?

Let $n,x\in\mathbb{N}$, $\alpha,\beta,\lambda\in\mathbb{R}^+$, where $\alpha,\beta<1$. Does the following sequence converge (and to what)? $s_0=\alpha n+\lambda$ ...
2
votes
1answer
25 views

Sums Involving the Mobius Function

Are there any good approximations for the following sums in terms of $n$? $$\sum_{k=1}^{n}\mu(k)$$ $$\sum_{k=1}^{n}\mu(k)\log^m(k)$$ $$\sum_{k=1}^{n}\frac{\mu(k)}{k}.$$ I realize that the third sum ...
-2
votes
1answer
48 views

$\sum r 2^r$ series computation [on hold]

How do you compute this series? $$ \sum_n n 2^n $$ I tried it the same way as computing the geometric series but I didn't come to a result Hope someone could help Edit The series goes from 0 to ...
0
votes
0answers
32 views

Sum from zero to -1 [duplicate]

I know that it does not make any sence, but I need it in order to prove something. I would like to ask, is it possible to have a sum which goes from zero to -1? I mean can this be defined normally and ...
-1
votes
1answer
42 views

Sum C(n,k)/k An easy expression? [on hold]

here is my problem : http://www.wolframalpha.com/input/?i=sum+from+k%3D+1+to+n+C%28n%2Ck%29%2Fk Though the result, i think wa can find a result with only n. Thanks !
0
votes
2answers
51 views

summation of ceil and floor function

I need a closed solution or a faster algorithm for calculating $$ \sum_{k=1}^{n-1} \left\lceil \frac{n}{k}-1 \right\rceil $$ and $$ \sum_{k=1}^{n-1} \left\lfloor \frac{n}{k} \right\rfloor $$ where $ ...
0
votes
1answer
112 views

Meaning of $\sum_{4}^{0}$

Encountered this question in Knuth et al's concrete mathematics. The question is what does $\sum_4^0$ mean? I think it does not mean anything unless we assign it meaning. It's just notation that can ...
1
vote
1answer
23 views

Factorials in Sigma Notation

Geometric Series one would use $S_n = \dfrac{a_1\cdot (1 - r^n)}{(1 -r)}$. Arithmetic Series one would use $S_n = \dfrac{n\cdot (a_1 + a_n)}{2}$. But how would I convert a sigma notation problem with ...
0
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2answers
23 views

Approximating the sum of integers with the logarithm

Why does the following hold? $\sum_{j=1}^{n-1}j \to \log(n) \text{ as } n \to \infty$ Thanks!
4
votes
5answers
86 views

How can I find integers $n \gt 1$ such that the average of $1^2,2^2,3^2…n^2$ is itself a perfect square.

$\sum_{i=1}^n i^2=\frac{n(n+1)(2n+1)}{6}$ so we would like to solve $6k^2=(n+1)(2n+1)$ here we see that $6|(n+1)(2n+1)\implies 2|n+1$ hence we can set $n=2j-1$ $2j(4j-1)=6k^2\implies ...
2
votes
2answers
32 views

How to get sum of points along curve?

startingQuantity / numberAvailable = price 5 apples / 4 available = $1.25 So if there are 4 available and the price is 1.25 and someone buys three apples then ...
0
votes
1answer
49 views

A formula for $\sum^n_{i=1}(1+1/n)$?

Find a formula for $$\sum^n_{i=1}\left(1 + \dfrac{1}{n}\right)$$ Prove that it holds for all $n \geq 1$. It kind of looks like is a series but I didn't succeed in this problem. Can someone help me ...
2
votes
3answers
48 views

Is there a simplification of this sum of quadratic sums?

I have $a(1+2+3+...+N)^2+a(2+3+...+N)^2+a(3+..+N)^2+...+aN^2$, essentially a sum of quadratic terms holding an integer summation that get consecutively smaller Within each term the summation up to N ...
1
vote
0answers
28 views

Extracting coefficients from a transformed generating function

Let $G(z)=\sum_{k\geq 0} a_kz^k$ be a generating function such that $z^aG(1-z)=P(z)$, where $P(z)$ is a polynomial and $a$ is a positive integer. I'm interested in $P(z)[z^n]$, the coefficient in ...
4
votes
3answers
178 views

How to closed the sum $\displaystyle \sum_{k=0}^n \dfrac{(-1)^k(2k+1)!!}{(n-k)!k!(k+1)!}$

How to closed the sum $\displaystyle S=\sum_{k=0}^n \dfrac{(-1)^k(2k+1)!!}{(n-k)!k!(k+1)!}$ I'm trying divide two cases $n$ odd and $n$ even. I predict that ...
1
vote
1answer
18 views

Convert sum to function

I need to convert $\sum_{i=0}^N \frac{C_1}{C_2+C_3i}$, to a function $C_1$, $C_2$ and $C_3$ are constants. I am interested in resulting function itself and method as well.
0
votes
1answer
48 views

How do I solve this summation [duplicate]

$$ \sum_{b=1}^{n} \lfloor\frac{n}{b} \rfloor $$ I can't figure out how to convert this into a closed form. Please help Thanks! Edit: I often come across summations I'm unable to solve. Is there ...
1
vote
1answer
47 views

Bound summation of successive square roots

What is a tight upper bound for $f(n)$ where $f(n) = f(\sqrt{n}) + \frac{1}{n}$. One can easily find the following upper bound $O(\lg \lg n)$, however I'm interested in a tight bound. Regards.
0
votes
3answers
42 views

Finding the Formula For the Sum of a Sequence

In the problem below, It is asked to find the formula for the sum of the sequence and then to prove whether it is true or false for all n values using induction. $$ 1 + 4 + 7 + ... + (3n + 1), \ n\in ...
0
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0answers
23 views

Is there an upper limit to the number of times a value can occur in a superset?

Given a set of numbers S=(-5,6,9,3,2,-2,), is there an upper limit to the number of times a particular value (say 4) can occur in the sums of all the combinations of these numbers? For example: in ...
0
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0answers
55 views

Is this proof correct?

I was working on a summation problem, and I thought of a way to solve it. This is a proof of a generalisation of that method, is it correct? Let $[a,b]_n$ denote the set of a partition of the ...
0
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0answers
35 views

Precise notation of a sum of a sequence

I need help in rewriting the support of a function $f$ in a more compact or precise way given its upper bound $b$ and lower bound $a$ as \begin{eqnarray} b&=&\max\left( \sum_{n}\alpha_n ...
0
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2answers
33 views

How can i resolve this equation?

Consider the following property $ P(n) $: $ \sum_{k=1}^{n} k = \frac{1}{8}(2n+1)^2 $ Show that $\forall n (P(n) \Longrightarrow P(n+1))$ Where do i start?
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0answers
27 views

Sum with conflicting specifier

This is the fomula (there might be some terms added, it is just boiled down to these two sums): $$\sum_{i = 0}^{N} \Big(\ Term_i * \sum_{j = 0}^{i - 1} x_j\ \Big)$$ The outer sum will iterate from $0$ ...
1
vote
1answer
35 views

Problem Relating to Error in Series

For the following series, find the number of terms required to find the sum with error < 0.005, and find upper and lower bounds for the sum using a much smaller number of terms. ...
5
votes
3answers
656 views

Preventing “proof by homework”?

I am doing problem 3d in the Prologue of Spivak: Prove $(a+b)^n = a^n + {n\choose1}a^{n-1}b + {n\choose2}a^{n-2}b^2 + ... + {n\choose n-1}ab^{n-1} + b^n$ I feel like my proof could pass off as ...
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0answers
21 views

Find Sum of series

Find Sum of series - $$ C^n_k + C^n_{k+t} + C^n_{k+2t} + C^n_{k+3t} + ... C^n_{k+qt} $$ Here $$ k+q\cdot t \le n $$ $$ q\ge 0 $$ $$ k \le n $$
4
votes
0answers
76 views

How to prove this indentity $\binom{100}{0}^2-\binom{100}{1}^2+\binom{100}{2}^2-…-\binom{100}{99}^2+\binom{100}{100}^2=\binom{100}{50}$ [duplicate]

I don't know how to prove this identity: $\binom{100}{0}^2-\binom{100}{1}^2+\binom{100}{2}^2-\binom{100}{3}^2+...-\binom{100}{99}^2+\binom{100}{100}^2=\binom{100}{50}$
0
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2answers
27 views

Help with telescoping sum

How can I use the telescoping technique to compute the following sum? I'm having issues getting started. I know the basic steps but I don't know how to perform them. I know I have to separate the ...
0
votes
2answers
26 views

How can we prove this = 1 for all n

$\displaystyle n!-\sum_{k=1}^{n-1}k\cdot k!$ By computing this by hand for several small values of $n$ I can see that it is always equal to 1. But I can't see how to prove that.
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0answers
16 views

How to solve this sum of exponents times polynomials?

$\sum^{\infty}_{n=0}\exp[-an](n^2-1)\frac{x^n}{\sum^n_{m-0}x^m}$= ? , where $0<x<1$ and $a$ is a variable. Can we get an analytic answer?
0
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1answer
37 views

Compute the following sum

I am to compute the following sum and my professor wrote this on the board. Although I can see what he is doing here and how to use the S and 2S I can't figure out the steps that are highlighted in ...
0
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0answers
33 views

How can this equation be solved with respect to mu?

How can this equation be solved with respect to mu? I'm not even sure whether this equation can be solved or not. If it's not possible, could you explain why?
1
vote
2answers
20 views

Summation Value of Combination Function

$$\sum_{n=1}^{\infty}\frac{5^n}{n!2^n}$$ I know how to prove that the above series is convergent but I'm not sure how to get the value that it converges to. If it were just the geometric series, I ...
0
votes
0answers
40 views

Is there a closed form of the following expression.

Does anyone know of a way to write a closed form of the following expression using only addition, subtraction, multiplication, and division? $$ \left[\sum\limits_{i=1}^n \ln(i)\right]^e $$
1
vote
3answers
51 views

How do these two summations equate?

Apparently, the summation $$ \sum_{j = i + 1}^n \frac{1}{j - i + 1} $$ is equal to the summation $$ \sum_{k=1}^{n - i} \frac{1}{k + 1} $$ I don't grasp the intuition behind why.
0
votes
1answer
44 views

closed form for $\sum_{k=0}^{n-p}\binom{n}{k}\binom{n}{p+k}$

how to get closed form for $$\sum_{k=0}^{n-p}\binom{n}{k}\binom{n}{p+k}$$ I tried to write binominal in term of gamma function but I got no result what is your suggest to solve the problem ?
0
votes
0answers
17 views

Sum rules for group-cohomology cocycles?

Consider a cocycle ($\omega$) in $H^n(G,U(1))$, where $U(1)$ is the group of unitary matrices over ${\rm C}$ of dimension 1. Thinking of $U(1)$ as a circle in the complex plane, one can multiply and ...