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

learn more… | top users | synonyms

0
votes
2answers
15 views

Summation with non integer induces

All of the sums I've encountered so far have been functions $f:Z\rightarrow$ (any other set). Or in other words the sums are in the form of $\sum_{j\in Z\lor j\in N}^n a_j=a_0+a_1+...+a_n$. This ...
2
votes
4answers
46 views

How to calculate the following sums?

I would like to know of a way to evaluate the following two for arbitrary $n$. $$\sum_{i=1}^ni!\,, \quad \sum_{i=1}^n \frac{n!}{i!}. $$
1
vote
1answer
12 views

help with simplifying this sum

Problem I need help with simplifying following sum: $$ 1 + \sum_{i=1}^{\infty}{\frac{1}{i!} * (-1)^i * a * (a + b)^{i-1}} $$ and can get the $a$ out to get $$ 1 + ...
0
votes
2answers
33 views

Sum of this eries: $\sum_{k=1}^{\infty}kp(1-p)^{k-1}$

$\sum_{k=1}^{\infty}kp(1-p)^{k-1}$ Can someone help me evaluate this sum? I couldn't even start, I have just written down the first couple of elements, but didn't help either. Thanks!
1
vote
0answers
36 views

Turning a summation into an integral

I have a summation of the form: $$y(x) = \sum\limits_{h=-L}^L\frac{A(h)\cdot R(h)^2}{((x-h)^2+R(h)^2)^{3/2}}$$ Where I wish to solve/optimise $R(h)$ (leaving $A(h) = const/h$) or $R(h)$ and $A(h)$ ...
0
votes
2answers
20 views

Find a number in a set that is not equal to sum of any other numbers

Given a set of nonzero real number S= $\{d_1,\cdots, d_N\}$, can I find a number $d_e \in S$ such that there does not exist $d_m,d_j\in S$ such that $d_e = d_m+d_j$ ? Assume $m$ and $j$ can be equal ...
0
votes
1answer
9 views

Sum of edge numbers for triangle given starting number, increment and number of levels

For example, if starting number (N) = 1, increment (I) = 5, and number of levels (L) = 4, you get the following triangle: 16 11 11 6 6 1 1 ...
2
votes
0answers
24 views

Lower bound of $\sum_{k = 1}^{N}1/(x + k)$

Let $f(x) := \sum_{k = 1}^{N}1/|x + k|$ for $x \in [0, N]$. Why is $f(x) \geq C\log N$ for all $x \in [0, N]$ where $C$ is an absolute constant. My work is: Since $x \in [0, N]$, we can remove the ...
1
vote
3answers
40 views

Question about a sum

Why is it that $$\sum_{k=1}^n(n+k+1)(n+1)=\frac{3}{2}\sum_{k=1}^n3k^2+k.$$ I cannot understand it. This is not homework, I am just a little interested in this!
0
votes
0answers
9 views

Proving a disequality involving hypergeometric distribution

How would you prove symbolically the following property? $$H(m_a+1,p_a+1,m,p) < H(m_a,p_a,m,p)$$ where $H(m_a,p_a,m,p)$ is the probability of drawing $m_a$ white balls in a series of $m$ ...
1
vote
1answer
27 views

How to find an asymptotic formula for $f(n)=\sum_{k=1620}^{n}(\log\log\log k)^{2}$?

How to find an asymptotic formula for function given below. $$f(n)=\sum_{k=1620}^{n}(\log\log\log k)^{2}$$
1
vote
0answers
29 views

Interchange of limiting operations (question from an engineer)

I need to clarify when are the below operations valid. If possible, please link me to the related theorems, where I can find details. 1- Given a double integral \begin{equation} \int_{X}\int_Y ...
5
votes
1answer
61 views

$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{(-1)^m E_{2m}\pi^{2m+1}}{4^{m+1}(2m)!}$

I'm looking for a way to prove $$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{(-1)^m E_{2m}\pi^{2m+1}}{4^{m+1}(2m)!}$$ Since ...
0
votes
2answers
27 views

Intervel of Convergence of a Power Series

Can anyone explain how to do this problem? I think you might be able to approach it with the ration test but I'm unsure. Any help is greatly appreciated! $$\sum_{n=0}^{\infty} \frac{(2x-3)^n}{n \ ...
3
votes
1answer
54 views

Number of palindromic numbers less than a power of $10$

I noticed that every $10^{n}$ there is a certain number of palindromic numbers that I collected in this sequence: $$S=\{a_n,a_{n+1},a_{n+2}...\}=\{10,9,90,90,900,900...\}$$ where every number $a_n$ is ...
2
votes
1answer
48 views

$\sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}$

Hi I am trying to calculate the sum given by $$ \sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}=\ = \sqrt{\frac{\pi}{\alpha}} e^{\beta^2/(4\alpha)} ...
2
votes
0answers
33 views

Finding sum for function with floor-function

I am trying to find a formula to calculate the following sum: $$\sum_{x=0}^n (2ax - {1 \over 2}a^2 - {1 \over 2} a) $$ where $$ a = \left\lfloor {x \over \phi^2} \right\rfloor $$ and $$\phi = {1 + ...
1
vote
1answer
38 views

Prove Convergence or Divergence

I just need to prove either convergence or divergence for this. Having some serious trouble and would appreciate all help! $$\sum_{n=1}^{\infty}\frac1{n^{1/3}(1+n^{1/2})}$$
2
votes
1answer
45 views

What is this sequence of polynomials?

NovaDenizen says the polynomial sequence i wanted to know about has these two recurrence relations (1) $p_n(x+1) = \sum_{i=0}^{n} (x+1)^{n-i}p_i(x)$ (2) $p_{n+1}(x) = \sum_{i=1}^{x} ip_n(i)$ == i ...
-1
votes
1answer
23 views

Explanation for sum of sequence

I saw that in a textbook. Could somebody explain how this sum of a sequence was obtained? ⌈n/2⌉+...+⌈n/2⌉+⌈n/2⌉ = ⌈(n+1)/2⌉⌈n/2⌉
0
votes
2answers
65 views

What is the name of this summation formula?

So recently I derived a formula (obviously not the first... it already existed but that is what got me into summations) that quickly adds all the numbers from 1 to "n" However I recently derived ...
0
votes
3answers
54 views

How can one determine whether the following series converges or diverges [duplicate]

$$ \sum_{n = 2}^{\infty}\frac{(-1)^{n}}{\sqrt{n} + (-1)^{n}} $$ Wolfram Alpha returns nothing useful, except that the ratio test was inconclusive.
0
votes
0answers
14 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
58 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
95 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
41 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
23 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
16 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
971 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
66 views

Question on the Prime Number Theorem (the Tchebychev Function) [duplicate]

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
119 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
50 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
60 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
55 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
28 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 [closed]

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
33 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
45 views

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

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
55 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
113 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
26 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
votes
2answers
24 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
89 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
49 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
29 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
183 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.