Tagged Questions

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

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1
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0answers
5 views

A question from the dreams realm

Let $\phi:\mathbb{R}\rightarrow\mathbb{R}$ be a function. Let $\phi_0(x)=\phi(x)$ and $\forall k\in\mathbb{N},\phi_{k+1}(x)=\phi(x\cdot\phi_k(x))$. Let $k\in\mathbb{N}^*$. What can be said of the ...
0
votes
1answer
15 views

sum of two equal digit numbers vs. sum of those digits

if I take 5688+6984=12672 then sum the result 1+2+6+7+2=18 then sum that result 1+8=9. vs. this. same digits from above. 5+6+8+8+6+9+8+4=54 then sum that result 5+4=9. using this method where the ...
1
vote
1answer
11 views

Intersection of two lines and the minimum of the sum of the two.

We use a formula in my Operations Research class for finding the 'Economic Order Quantity', given the cost function (sum of Holding and Ordering costs) $$C = \frac{Q}{2}H+\frac{D}{Q}S$$ where $Q$ is ...
2
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1answer
52 views

Finding the infinite sum of $e^{-n}$ using integrals

I am trying to understand this: $\displaystyle \sum_{n=1}^{\infty} e^{-n}$ using integrals, what I have though: $= \displaystyle \lim_{m\to\infty} \sum_{n=1}^{m} e^{-n}$ $= \displaystyle ...
6
votes
3answers
56 views

Find $S=\frac{1}{2}+\frac{3}{2^2}+\frac{5}{2^3}+\frac{7}{2^4}+…+\frac{2n-1}{2^n}+…$

I'm trying to calculate $S$ where $$S=\frac{1}{2}+\frac{3}{2^2}+\frac{5}{2^3}+\frac{7}{2^4}+...+\frac{2n-1}{2^n}+...$$ I know that the answer is $3$, and I also know "the idea" of how to get to the ...
2
votes
2answers
17 views

Equivalence of summations

Show that $$\frac{1}{n}\sum^{n}_{i=1} (x_{i} - \bar{x})^{2}\equiv \frac{1}{n}\sum^{n}_{i=1}x_{i}^{2} - \bar{x}^{2}.$$ Note that $\bar{x} = \frac{1}{n}\sum^{n}_{i=1} x_{i}$. So I have started by: ...
0
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2answers
40 views

find the sum of series

I have problem with finding sum of series: 1)$\displaystyle\sum_{n=k}^{2k-1}\frac{n}{2^n}=?$ 2) $\displaystyle\sum_{n=0}^{k-1}n(\frac{4}{3})^n=?$ I have some idea to 1) ot write it as ...
2
votes
1answer
23 views

Integration of $x^a$ and Summation of first $n$ $a$th powers

I'm learning some discrete mathematics. I already knew a little (very little) calculus, and I noticed something. I think it's just a coincidence, so I'm sorry if this is a bad question. There are some ...
5
votes
1answer
38 views

How to prove $\sum_{k=1}^{N} \frac{\sin n\theta}{2^N}=\frac{2^{N+1}\sin \theta + \sin N\theta -2\sin(N+1)\theta}{2^N(5-4\cos \theta)}$

Prove This using De Moivre Theorem $$\sum_{n=1}^{N}\frac{\sin n\theta}{2^n}=\frac{2^{N+1}\sin\theta+\sin N\theta-2\sin(N+1)\theta}{2^N(5-4\cos\theta)}$$ Please help me find my mistake, because ...
4
votes
2answers
13 views

Control ratio of geometric series through its sum

A geometric series $S_n$ is the sum of the $n$ first elements of a geometric sequence $u_n$: $$u_n = ar^n \space \forall n \in \mathbb{N}^*$$ with $u_0$ defined, and: $$S_n = \sum_{k = 0}^{k = n - ...
0
votes
1answer
17 views

Plotting discrete time signals involving sumations in matlab.

Many of the examples I've encountered while looking for an answer are simple functions that do not involve summations. Suppose I have the following function; ...
3
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2answers
39 views

Is there a Sum Factorial?

I am curious if there is any addition factorial. Obviously, $$x! = \prod_{n=1}^x n$$ but what I want is a shorthand way of writing: $$\sum_{n=1}^x n$$ So is there such a thing? and if so, what is ...
3
votes
2answers
29 views

Converting Summation to Expression

How does the summation break down from $$\displaystyle\sum_{n \geq 0} (x + x^2) ^ n$$ to $$\frac1{1 - x - x^2} $$ per this answer?
0
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1answer
33 views

Using induction to prove that $ \prod_{i=1}^{n} (1+a_{i}) \geq 1 + \sum_{i=1}^{n}a_{i} $ [on hold]

I started a course in my university and I am having trouble with answering this question: Prove using Mathematical induction, for every real, non-negative 'n' number $$(a_{i}\geq 0)$$ the ...
0
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0answers
19 views

Simplifying Double Summation

$\sum_{k=1}^m\sum_{l=1}^n(4k-l+2) =4n\sum_{k=1}^mk-m\sum_{l=1}^nl+2mn$ We simplified this Equation to use the formula $\sum_{k=1}^nk$=$n*(n+1)\over2$ I don't understand where the $n$ in ...
1
vote
1answer
30 views

Summation of multiplicative function $f$ where $f(p) = 1$ for $p$ prime

I have a multiplicative function $f$ with a special "base" case: For every prime $p$, $f(p) = 1$. E.g. splitting up $f(3^5 \times7^2 \times 13 \times 17)$ yields $f(3^5) f(7^2)$ which is left to be ...
1
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3answers
25 views

Closed form of a sum

I am trying to derive the closed form of the sum $\sum\limits_{i=2}^n \frac{1}{i(i-1)}$ which Mathematica tells me is $\frac{n-1}{n}$. I am completely baffled on how to arrive at this result. The ...
2
votes
3answers
283 views

Evaluate the sum.

Evalute the following sum: $ 1 + 2 + 2 + 3 + 4 + 4 + 5 + 6 + 6 + . . . .+ (n-1) + n + n$. I tried doing it but I keep getting the wrong answer. I've used known sums to solve it.
3
votes
4answers
132 views

proving an invloved combinatorial identity

How to prove following Identity? $$\sum_{k=0}^n (-1)^k {n-k \choose k} m^k (m+1)^{n-2k} = \frac {m^{n+1}-1}{m-1}, m \ge 2$$ This seems very hard to me. Any idea about how to prove it combinatorialy? ...
4
votes
3answers
117 views

Replace a sum with an integral $\sum\rightarrow \int$

How can one turn a sum to an integral. Example $$\sum_k f(k) \approx N\cdot\int_k dk\, f(k). $$ How do you find the factor $N$? The quantities should be approximately equal. Example form Peskin ...
4
votes
2answers
48 views

Summing a series to $n^{2}$

I'm currently trying to to sum the following series: $$ \sum_{k=1}^{n^2}\frac{1}{1 + \left ( \frac{k}{n} \right)^{r}} $$ I'm not sure what to do since we are summing over $n^{2}$. Any help would be ...
2
votes
2answers
30 views

Geometric Series of $a = 2$, $r = 2$

I'm having some trouble determining the geometric series for the following: $$2^k + 2^{k-1} + \cdots + 2^2 + 2$$ With this, I'm guessing that $a = 2$, and common ratio is: $r = 2$ Most resources I ...
4
votes
0answers
26 views

Write $\sum_{k=1}^nk\sin(kx)^2$ in closed form

$\underline{Given:}$ Write in closed form $$\sum_{k=1}^nk\sin(kx)^2$$ using the fact that $$\sum_{k=1}^nku^k=\frac u{(1-u)^2}[(n)u^{n+1}(n+1)u^n+1]$$ $\underline{My\ Work:}$ I substituted ...
3
votes
2answers
100 views

How to prove that $\sum_{k=0}^n\cos(k\pi)\binom{n}{k}=0$?

$$\sum_{k=0}^n\cos(k\pi)\binom{n}{k}=0$$ I approached this problem having no idea that $\cos(k\pi)$ could be substituted so easily. I tried first to expand the sum to $n$. So I wanted to ask if ...
0
votes
4answers
39 views

why is $\sum_0^{n-1} k(k-1) = n(n-1)(n-2)/3$ [on hold]

how do I get that $\sum_0^{n-1} k(k-1) = n(n-1)(n-2)/3$? And why is it true?
3
votes
0answers
44 views

When does sum to infinity starts getting negative? [duplicate]

There is a popular claim going around the internet that the sum of the positive integers is $-1/12$. There are proofs to this statement and I am not going to try and refute them. But I could not ...
0
votes
3answers
18 views

Summation using cdf of binomial distribution

I'm trying to find the exact value of the following equation: $$\sum_{x=0}^{1000}\tfrac{x^{2}-x+5}{x!(1000-x)!}2^{x}7^{1000-x}$$ I've managed to convert to the following: ...
0
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0answers
21 views

How do I simplify $\sum_i^n(y_i-rx_i)^2$, where $r = \frac{\sum y_i}{\sum x_i}$?

I want to simplify: $$\sum_i^n(y_i-rx_i)^2$$ where $y_i$ and $x_i$ are random variables and $r = \frac{\sum y_i}{\sum x_i}$. I've tried expanding the summand and replacing $r$ with $\frac{\sum ...
0
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0answers
8 views

Sum of Inverse Partitioned Matrix

Given a matrix $X(n\times p)$, divide $X$ by row into $K$ parts, $X_1, X_2...X_K$ each of which consists of the same amount of row vectors in $X$ as its own row vectors. Now consider ...
0
votes
2answers
29 views

How to find the value of $\sum_{k=1}^\infty (\frac{1}{9})^k$ using partial sums?

So I was trying to prove an infinite sum by looking at the partial sum, when I ran into a problem. Consider: $$\sum_{k=1}^n \left(\frac{1}{9}\right)^k = \frac{1}{8} 9^{-n} (9^n-1)$$ but as there are ...
2
votes
3answers
119 views

Is the limit finite? (corrected)

I need to find $r>0$ for which the following limit is finite $$\lim_{n \rightarrow \infty} \sum_{k=1}^{n^2} \frac{n^{r-1}}{n^r+k^r}$$ I get inconclusiveness using the ratio test. The root test ...
1
vote
3answers
51 views

Is there any way to simplify following summation?

Is there any way to simplify following summation? $$\sum_{k=1}^n \frac{1}{k^2(k+1)^2}$$
2
votes
2answers
37 views

Finding a number $M$ such that $S_N<10^{-20}$ for all $N>M$

Given that $$u_n=\frac{1}{n^2-n+1} - \frac{1}{n^2+n+1}$$ $$S_n=\sum_{n=N+1}^{2N} u_n$$ Find a number $M$ such that $S_N<10^{-20}$ for all $N>M$ I did: $$S_N = ...
1
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1answer
53 views

Prove that $\sum_{i=1}^na_i\sum_{i=1}^na^{-1}_i\ge n^2$ and $\sum_{i=1}^na_i^2\ge\frac1n$ [on hold]

For $a_i>0$, $i=1, \dots,n$ prove the inequalities $a)$ $$\sum_{i=1}^na_i\sum_{i=1}^na^{-1}_i\ge n^2$$ $b)$ $$\sum_{i=1}^na_i^2\ge\frac1n,\quad \text{if additionally}\sum^n_{i=1}a_i=1$$ ...
2
votes
2answers
41 views

Looking for a closed form for $\sum_{k=1}^{\infty}\left( \zeta(2k)-\beta(2k)\right)$

For some time I've been playing with this kind of sums, for example I was able to find that $$ \frac{\pi}{2}=1+2\sum_{k=1}^{\infty}\left( \zeta(2k+1)-\beta(2k+1)\right) $$ where $$ ...
1
vote
1answer
157 views

How is this summation solved?

$$\sum_{i=\left\lceil\frac{n}{2}\right\rceil}^n \left\lceil\frac{n}{2}\right\rceil^k$$ This summation is a part of a proof (asymptotic lower bound) I was reading in my textbook, but I don't ...
11
votes
5answers
139 views

Finding $\sum \frac{1}{n^2+7n+9}$

How do we prove that $$\sum_{n=0}^{\infty} \dfrac{1}{n^2+7n+9}=1+\dfrac{\pi}{\sqrt {13}}\tan\left(\dfrac{\sqrt{13}\pi}{2}\right)$$ I tried partial fraction decomposition, but it didn't work out after ...
1
vote
1answer
29 views

How to Change Summation Expression $\sum_{i=1}^N \mathbf{X}_i^{\top}\mathbf{\Omega}^{-1}\mathbf{X}_i$ into Matrix Expression

Let $\mathbf{X}_i$ be a $G \times K$ matrix, and suppose are $i=1,...,N$ of these matrices. Note that \begin{align} \sum_{i=1}^N \mathbf{X}_i^{\top}\mathbf{X}_i &= \begin{bmatrix} ...
1
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0answers
33 views

Can I do the following when solving my integration??

I appreciate any feedback for my question. I have an integration as follows $$\int_{-\pi}^{\pi}\frac{1}{2\pi} \prod_i \frac{1}{1+ x_ig(\theta)} d\theta $$ I have that $g(\theta)$ is the defined as ...
-2
votes
3answers
65 views

The sum of all the odd numbers to infinity [duplicate]

We have this sequence: S1: 1+2+3+4+5+6.. (to infinity) It has been demonstrated, that S1 = -1/12. Now, what happens if i multiply by a factor of 2? S2: 2+4+6+8+10+12.... (to infinity). I have ...
1
vote
1answer
54 views

Search for summation formula

Is there any closed formula for the sum $ ~\sum_{k = 1} ^ {n} r^k k^r ~$ where $~r~$ is an integer? Thank you very much in advance.
0
votes
4answers
53 views

An infinite series question. [closed]

Ok, so we have an infinite sequence: S1 = 6+14+22+30+38.. Now, there is another infinite sequence, S2 = 1+2+3+4+5+6+7.. We know that the first sequence's nth term = (8n-2 ) so surely there must be a ...
1
vote
1answer
13 views

Calculate total revenue share

Suppose you have data such as this: January Total minutes of all videos watched: 50 Total minutes of video X watched: 25 Total revenue: 200 February Total minutes of all videos watched: 200 Total ...
8
votes
4answers
89 views

Finding the sum of $\sin(0^\circ) + \sin(1^\circ) + \sin(2^\circ) + \cdots +\sin(180^\circ)$

I need help understanding the sum of $\sin(0^\circ) + \sin(1^\circ) + \sin(2^\circ) + \cdots +\sin(180^\circ)$ or $\displaystyle \sum_{i=0}^{180} \sin(i)$ This might be related to a formula to find ...
0
votes
1answer
23 views

Find $\sum_{i=0}^{\log n} \frac{1}{2^i}$

I'm not really sure how to solve summations, so any help would be great. In particular, I had thought that $n^2\sum_{i=0}^{\log n} \frac{1}{2^i}=O(n^2\log n)$ but it's actually $O(n^2)$, and I'm ...
-1
votes
1answer
27 views

Set of a summation

Let $S = \{n ∈ N | n \text{ divides the sum of any n consecutive numbers} \}$. How can I describe the set S? I was given the hint: $\displaystyle\sum\limits_{n=1}^N n=\frac{N(N+1)}{2}$ I don't want ...
0
votes
0answers
6 views

Summation of all combination

I have two matrix. A=[1 2 3];B=[4 5 6]; the all possible combination of their summation is [1+4 1+5 1+6; 2+4 2+5 2+6;3+4 3+5 3+6]. Now instead of 1*3 my matrix dimension is 1*n. and instead of two I ...
2
votes
1answer
73 views

Compute $\sum_{k=0}^{n}\frac{1}{\binom{n}{k}}$

I want to calculate $\sum_{k=0}^{n}\frac{1}{\binom{n}{k}}$. No idea in my mind. Any help? Context I want to calculate the expected value of bits per symbols in adaptive arithmetic coding when the ...
0
votes
1answer
35 views

summation of series by telescoping series method (feedback needed)

i am stuck i did the first part by cancelling out terms since its a telescoping series. But I do not know how I can proceed any further . Please help. I am not sure of whatever i have done so far. so ...
1
vote
3answers
128 views

Find if this series converges and if so find its value

I need help I cant understand how we can solve this. I am confused when the log came in. I listed the first few terms but i do not know how to proceed further. all I know is that the sequence is ...