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

learn more… | top users | synonyms

3
votes
0answers
15 views

Sum of Complex Numbers and Modulus Inequality

Let $z_{1}, \dots, z_{n} \in \mathbb{C}$. Then, there exists a subset $S \subset \{1,\dots,n\}$ such that: $ \left| \displaystyle\sum_{j \in S}z_{j} \right| \geq ...
4
votes
0answers
33 views

A closed form for the following Series

I was computing some calculations, when I got stuck about a possible closed form for this series: $$S = \sum_{k = 2}^{N}\ \frac{k!}{k^k - k!}$$ I proved by hands that it's absolutely convergent by ...
1
vote
2answers
31 views

Simplify triangular sum of triangular numbers: $\sum_{i=1}^{n}(\frac12i(i+1))$

I'd like to simplify this expression, which sums up the first $n$ triangular numbers: $$\sum_{i=1}^{n}(\frac12i(i+1))$$ which is equal to: $$\sum_{i=0}^{n}((n-i)(i+1))$$ Is it even possible without ...
1
vote
2answers
63 views

Prove that $\frac 12\leq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}…+\frac{1}{n+n}$

How do I prove $$\frac 12\leq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}...+\frac{1}{n+n}$$ without using induction? Note that clearly $n\neq 0$ Thanks for any help!!
1
vote
1answer
38 views

Closed-form Solution of Log Sum

I have the series: $$\sum_{i=1}^{i=10^N} \log_5 i$$ I'm trying to figure out how to get the closed-form solution to this problem. I entered it into WolframAlpha and got that it equals: $ ...
-1
votes
2answers
103 views

What $+1+1+\cdots$ really equals

$1+1+1+\cdots$ is clearly a divergent series, so you'd say that it tends towards infinity? Through analytic continuation of the zetafunction the value $-1/2$ could be assigned the sum, right? But if ...
0
votes
5answers
58 views

Closed-form Solution to a Sum

I have some math questions for a programming course where it says to provide closed-form solutions for a list of sums. I've never taken an algorithms course, so I'm not quite sure what I'm doing. I ...
2
votes
2answers
47 views

Factorial Summation Definition

A while back I found the series $$\sum_{k=0}^n \binom n k (-1)^k (x+k)^n = (-1)^n n!$$ while messing around in Algebra class (specifically when $n$ is any natural number and $x$ is any real number) I ...
5
votes
1answer
86 views

How to solve this hard sum problem?

$$\sum _{ x=1 }^{ \infty }{ \frac { 3{ x }^{ 2 }+12x+16 }{ { \left( x\left( x+1 \right) \left( x+2 \right) \left( x+3 \right) \left( x+4 \right) \right) }^{ 3 } } } =\frac { 1 }{ 4{ (a!) }^{ b } } ...
1
vote
1answer
17 views

Function Equivalent to the Maximum Operator?

All numbers are real, WLOG positive. $A + B + ... + N = T$ and $A' + B' + ... + N' = T$ I'm trying to figure out some function, f, such that if $f(A,B,... ,N) > f(A',B',...,N')$ then, ...
-1
votes
0answers
43 views

Closed form for $\left(\sum_{k=0}^n\frac{x^k}{k!}\right)^p$

The expression for the p-th power of the sum of the first $n+1$ powers of x is given analytically by ...
5
votes
3answers
38 views

Combinatorial argument for $\sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i} = \binom{n}{i}2^{n-i}$

I need to show that $$\sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i} = \binom{n}{i}2^{n-i}$$ I know that $\displaystyle \binom{n}{k}\binom{k}{i}$ is counting the number of ways to pick $k$ elements ...
1
vote
0answers
26 views

Possible closed form or approximation?

Does it have some closed form or approximation ? I tried on my own but i am not getting any idea regarding this. $$\sum_{k_1=k}^{M}\sum_{k_2=k}^{M}\frac{k_1^{-\gamma} k_2^{-\gamma} ...
0
votes
1answer
21 views

how can $((\frac1N \sum z_i )- I )^2 = \frac1{N^2}(\sum (z_i - I)^2 )$?

$$\left(\left(\frac1N \sum z_i \right)- I \right)^2 = \frac1{N^2}\left(\sum (z_i - I)^2 \right)$$ How does this work? Or is there an error? I thought that you could not pull the sum out of the ...
2
votes
1answer
65 views

Summation from Right to Left changes Accuracy?

I was looking at The accuracy from left to right and that from right to left of the floating point arithmetic sums which was asking for the accuracy of $$\sum_{k=1}^n\frac{1}{k^2}$$ from right to ...
2
votes
1answer
50 views

Does $\sum_{i=1}^n \frac{1}{i^2}=O(\ln(n))$?

I was looking at Why $\sum_{i=1}^n \frac{1}{i} =\mathcal O(\ln(n))$?. And there it was proved that $$\sum_{i=1}^n \frac{1}{i} =\mathcal O(\ln(n))$$ My question is that does this also stand for ...
14
votes
3answers
2k views

What do $\{ceps_q\}_{q=0}^Q$ and $\{a_q\}_{q=1}^p$ mean?

As a programmer who hasn't had any higher mathematical training, I sometimes find mathematical equations described in books or online that I'd like to implement in my programs, but they have symbols ...
0
votes
1answer
34 views

Differentiating $- \sum_{n \in \mathbb{Z}^2} e^{i n \cdot \alpha}\int_0^E\frac{1}{4\pi t}\exp({\omega^2 t - \frac{|x - n - y|^2}{4t^2}})dt$ wrt $x$?

I have a formula for the Ewald method which can be used to speed up computations when working with periodic Green's functions. I will need to take the derivative of the function $G(x, y)$ with respect ...
0
votes
1answer
14 views

Difficulty simplifying nested sums with different variables

I'm trying to work out an algorithm analysis problem, and I'm having some difficulty determining how a jump is made between two steps in the answer. $$ \begin{align} ...
1
vote
1answer
24 views
9
votes
0answers
89 views

Find value to the summation : $\sum_{n =1}^\infty \dfrac 1 {5^{n+1}-5^n+1}$

$$\sum_{n = 1}^\infty \dfrac 1 {5^{n+1}-5^n+1}$$ I can factorize denominator to $4\times5^n+1$ to confirm the series does not diverge, But how do I calculate its actual sum? The series is not a ...
0
votes
2answers
18 views

Infinite convergent sum with central binomial coefficient over k

Given the following sum: $$0.5\cdot\sum\limits_{k=0}^\infty \frac{1}{k+1}\binom{2k}{k}\cdot(0.25)^{k}$$ I know that the sum is supposed to converge to $1$. How would I go about evaluating it to get ...
0
votes
1answer
41 views

Find the closed form for the double sum $ \sum_{1\leq j \leq k \leq n }3^k=\sum_{j=1}^n \sum_{j=k}^n 3^k$

Find the closed form for the double sum $$ \sum_{1\leq j \leq k \leq n }3^k$$ Here is my attempt: $$ \sum_{j=1}^n \sum_{j=k}^n 3^k $$ What should I do next to get the closed form? Please help me
2
votes
1answer
24 views

Easy way of seeing if swapping summation is ok? (Generating functional derivation of Bell numbers)

On page 21 of his book generatingfunctionology (available for free on the author's homepage), the author rearranges the summations in the following way: ...
0
votes
2answers
55 views

How to find total number of sum of consecutive number of $n$? [duplicate]

How many ways are there to write $n$ as the sum of consecutive positive integers? Example: $15$ has $3$ consecutive sums: $1+2+3+4+5=15$ $7+8=15$ $4+5+6=15$
-2
votes
1answer
45 views

Dealing with phi function property

If $n=2^kN$, where $N$ is odd, then $$\sum_{d\mid n}(-1)^{n/d}\phi(d)=\sum_{d\mid 2^{k-1}N}\phi(d)-\sum_{d\mid N}\phi(2^kd)$$ I have no idea how to seperate things inside the left side. In a ...
5
votes
1answer
57 views

Prove inequality $1 < \frac{1}{n} + \frac{1}{n+1} + \ldots + \frac{1}{3n-1} < 2$

Prove the inequality $1 < \frac{1}{n} + \frac{1}{n+1} + \ldots + \frac{1}{3n-1} < 2$ For all $n \in \mathbb{N}$ I've done the right hand side, but can't do the left side of the inequality. For ...
2
votes
0answers
20 views

summation combinatoric again with floor function

$\sum_{n=1}^{33}\binom{3n}{\left \lfloor 1.5n-0.5 \right \rfloor}= ...$ $\binom{3}{\left \lfloor 1 \right \rfloor}+\binom{6}{\left \lfloor 2.5 \right \rfloor}+\binom{9}{\left \lfloor 4 \right ...
0
votes
1answer
38 views

Show that $\sum _ {i=1} ^{\lg n - 1} \frac 1 {\lg n - i} = \sum _{i=1} ^{\lg n - 1} \frac 1 i$ [closed]

I couldn't understand this summation: $$\sum\limits_{i=1}^{\lg n - 1} \frac{1}{\lg n -i} = \sum\limits_{i=1}^{\lg n - 1} \frac{1}{i} .$$ How did author transform LHS to RHS? Can you describe in ...
0
votes
1answer
34 views

Simultaneous equation with summation and square - how to solve?

$\mathbf{p}$ is a vector with dimension: $x \times 1$ $\mathbf{d}$ is a vector with dimension: $1 \times y$ $\mathbf{V}$ is a matrix with dimension: $x \times y$ $y \geq x$ $\mathbf{d}$ and ...
3
votes
1answer
28 views

Proving that maximizing a sum of functions of different independent variables is equivalent to maximizing each function

Let $$ \pi = f_1(x_1) + f_2(x_2) + f_3(x_3) + \dots + f_n(x_n) = \sum_{i=1}^n f_n(x_i) $$ where $f_i$ denote different functions and $x_i$ denote different independent variables Would proving that ...
0
votes
0answers
11 views

Showing summation of disjoint sets can be split

Take three sets $X,Y,Z \subseteq V$ $$\sum_{u\in X\cup Y}\sum_{v\in Z}f(u,v)=\sum_{u\in X}\sum_{v\in Z}f(u,v) + \sum_{u\in Y}\sum_{v\in Z}f(u,v) \text{ if $X \cap Y=\varnothing$}$$ It seems ...
2
votes
0answers
29 views

Factorial ratio sum of finite series

Given: $ S = \sum_{i=1}^{n-1}{i! \over n!} $ How would I find the sum for an arbitrarily large $n$ ? Example: $n=5$ $ S = \frac{1!}{5!} + \frac{2!}{5!} + \frac{3!}{5!} + \frac{4!}{5!} = 0.275 $
0
votes
1answer
26 views

Non-infinite geometric sum; does not start at 0 or 1

It's bee a long time since I've worked with sums and series, so even simple examples like this one are giving me trouble: $\sum_{i=4}^N \left(5\right)^i$ Can I get some guidance on series like this? ...
1
vote
1answer
31 views

Solving $\max_{x\in\prod_{i=1}^n s_i} \sum_{i=1}^n f(x_i)$ by maximizing for each $i$ individually.

First, I will clarify some of the notation: $$ x_i \in S_i,\; i\in \{1,2,\dots, n\} \quad x\in S, \quad S\equiv \prod_{i=1}^nS_i \text{ (direct product set)} $$ So basically, we have $x \in S$ which ...
1
vote
2answers
44 views

Find the generalized sum of $1+2(2)+3(2)^2+4(2^3)+…+n(2^{n-1})$ [duplicate]

Find the generalized sum of $1+2(2)+3(2)^2+4(2^3)+...+n(2^{n-1})$ I rewrote the above sequence into: $\sum_{k=1}^{n} k(2^{k-1})$. The sequence looks like a hybrid of the summation $\sum_{k=1}^{n} ...
204
votes
8answers
12k views

The length of toilet roll

Fun with Math time. My mom gave me a roll of toilet paper to put it in the bathroom, and looking at it I immediately wondered about this: is it possible, through very simple math, to calculate (with ...
0
votes
2answers
42 views

Evaluating Nested Summations

I'm trying to evaluate the following nested summation as a function of $n$: $$\sum_{i=1}^{n-1} \sum_{j=i+1}^n \sum_{k=1}^j 1$$ So far I have: $$\sum_{i=1}^{n-1}\sum_{j=i+1}^n i+1$$ ...
1
vote
2answers
41 views

Limit of power series with L'Hospital

Calculate the given limit: $$\lim_{x\to 0} \frac{1}{1-\cos(x^2)}\sum_{n=4}^\infty\ n^5x^n$$ First, I used Taylor Expansion (near $x=0$): $$1-\cos(x^2)\approx 0.5x^4$$ I'm now quite stuck with the ...
0
votes
0answers
30 views

Number of ways a dice can roll every side equally many times for the first time after x rolls

This problem is best viewed as a walk on a $d$-dimensional integer lattice with integer steps corresponding to various results of a dice roll. For a normal 6-sided dice, these would be ...
1
vote
3answers
51 views

Find $\lim\limits_{n\to\infty}\left(\frac{1}{\sqrt{n}}\sum\limits_{k=1}^{n}\frac{1}{\sqrt{2k}+\sqrt{2k+2}}\right)$

I don't know how to find the sum of $\sum\limits_{k=1}^{n}\frac{1}{\sqrt{2k}+\sqrt{2k+2}}$. After rationalization we have ...
0
votes
1answer
48 views

Binominal expression simplification

I need to simplify the expression $$\sum_{k = 1}^{10} k\binom{10}{k}\binom{20}{10 - k}$$ Thank you.
7
votes
2answers
103 views

Find the sum $\sum _{ k=1 }^{ 100 }{ \frac { k\cdot k! }{ { 100 }^{ k } } } \binom{100}{k}$

Find the sum $$\sum _{ k=1 }^{ 100 }{ \frac { k\cdot k! }{ { 100 }^{ k } } } \binom{100}{k}$$ When I asked my teacher How can I solve this question . He responded it is very hard you can't solve it ...
3
votes
2answers
60 views

How to simplify $\lim_{n\to \infty}\sum_{r=1}^n \tan^{-1} \dfrac{2r+1}{r^4+2r^3+r^2+1}$

$$\lim_{n\to \infty}\sum_{r=1}^n \tan^{-1} \dfrac{2r+1}{r^4+2r^3+r^2+1}$$ How am I supposed to do it? One thing I see here is $$\lim_{n\to \infty}\sum_{r=1}^n \tan^{-1} \dfrac{2r+1}{(r^2+r)^2+1}$$ ...
4
votes
3answers
68 views

Find the sum of the $\sum_{m=k}^{+\infty}\binom{m}{k}(1-p)^k\cdot p^{m-k}$

Let $0<p<1$,Find the sum $$\sum_{m=k}^{+\infty}\binom{m}{k}(1-p)^k\cdot p^{m-k}$$
1
vote
1answer
30 views

How to neatly summarize indexes of a matrix where there are a lot of i's x j's [closed]

As you can see from the subject line, I can't even think of the word of what I need to do. I am trying to write in text that I multiplied columns of a matrix (n columns, i = 1:n). There are many ...
-1
votes
0answers
29 views

Two functions of cosines and stuff - Why aren't they equal for all $m$? [closed]

Please take look at this graph on Desmos. I didn't want to retype everything. My question is, why are $F$ and $G$ not the same for larger values of $m$? I worked out that the two functions should be ...
3
votes
4answers
87 views

Converting Σ i(i + 1) into a formula, given this hint

The given summation is: $$\sum_{i=1}^n i(i+1)$$ The goal is to convert it into a formula which only uses n. Solving this, I got the answer: $$\frac{n}{3}(n+1)(n+2)$$ However, I don't believe the ...
0
votes
0answers
30 views

Strange use of sigma notation in computability

Ok everyone, so I was reading about computability when I came across the following- ''Suppose that $f(x, z)$ is any function; the bounded sum $\sum_{z<y} f(x, z)$ is a function of $x, y$ given by ...
0
votes
1answer
19 views

What are the steps in between to derive at the solution of this summation? [closed]

What are the steps in between to derive at the solution of this summation?