Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.
0
votes
1answer
42 views
Questions concerning $\Omega$ and $\Theta$
How do I solve this: $\displaystyle\sum_{k = 1}^n \frac{k}{3}$ is $\Omega(n^2)$? I know that the summation would be $\displaystyle\sum\limits_{k=1}^n \dfrac{n^2+n}{6}$,
but how do I solve ...
2
votes
1answer
38 views
Convergent series and of positive integers and partial sums.
Let $\sum a_n$ be a convergent series of positive real numbers with sum $s$ and partial sums $s_n=a_1+a_2+\cdots+a_n$.
Prove that $\sum na_n$ is convergent if and only if $\sum (s-s_n)$ is ...
1
vote
1answer
42 views
$\sum_{i=1}^{n} (3i + 2n)$
I want to verify what would be the simplified solved version of this summation.
$$\sum_{i=1}^{n} (3i + 2n)$$
Would it be this?
$$ \frac32n^2 + \frac32n + 2n^2 $$
0
votes
0answers
99 views
A recurrence relation for Stirling numbers (2nd kind)
It is well-known that the Stirling numbers of the second kind satisfy the following (vertical) recurrence relation:
$$\sum\limits_{r=k}^n \binom{n}{r}S\left( r,k\right) =S\left( n+1,k+1\right) $$
...
2
votes
0answers
53 views
Pi identity with sum and product
Please prove this identity
$$\sum_{ n=1 }^{\infty }\left({\left(-1\right) }^{ n }\frac{\prod_{ j=1 }^{ n }{\left(\frac{ 3 }{ 2 }-j\right) }}{\left( 2n+1\right)\left( n!\right) }\right) =\frac{\pi }{ ...
3
votes
2answers
74 views
Summation/Sigma notation
There are lots of variants in the notation for summation. For example, $$\sum_{k=1}^{n} f(k), \qquad \sum_{p \text{ prime}} \frac{1}{p}, \qquad \sum_{\sigma \in S_n} (\operatorname{sgn} \sigma) a_{1 , ...
3
votes
2answers
167 views
Closed form sum of $\sum^{\infty}_{n=1} \frac{1}{3^n-1}$
Wolframalpha uses $q$-Polygamma function to represent the sum, hence essentially does nothing. Here I wonder if this sum can be represented by elementary function.
The summation is like a infinite ...
2
votes
2answers
80 views
Combinatorial Sum
I am trying to prove
$$0^2 \binom{n}{0}+3^2\binom{n}{3}+6^2\binom{n}{6}+ \cdots + \left[\dfrac{n}{3}\right]^2 \binom{n}{\left[\dfrac{n}{3}\right]},$$ where $[x]$ is the greatest integer not exceeding ...
3
votes
4answers
48 views
Solution to the geometric progression starting from an arbitrary index?
We know that:
$$
\sum_{n=0}^{n-1}r^n = \frac{r^n -1}{r-1}
$$
What about when the starting index is an arbitrary $n_0$? Is the following correct?:
$$
\sum_{n=n_0}^{n-1}r^n = ...
4
votes
2answers
54 views
Suppose $|\alpha_1| \le |\alpha_2| \le \cdots \le 1$, $n(r) = \#\{\alpha_j \le r\}$. Prove $\int_0^1n(r)dr = \sum_{j=1}^\infty(1-|\alpha_j|)$.
I'm trying to solve the following exercise from chapter 15 of Rudin's Real and Complex Analysis:
Suppose $|\alpha_1| \le |\alpha_2| \le \cdots \le 1$, and let $n(r)$ be the number of terms in the ...
0
votes
1answer
40 views
Handling summations with two variables
If I have a summation with let's say $x=0 \dots 500$ and $y=0\dots1500$
$500 \choose x$ $ 1500 \choose y$ $\dfrac{1}{2^{500}}\dfrac{2^{1500-y}}{3^{1500}}$,
How would I handle the constant? If I ...
2
votes
2answers
113 views
Exponential function formula proof
How does one arrive at $e^4$ from
$$\sum_{x=0}^{\infty}\frac{ 4^x}{x!}$$
3
votes
2answers
148 views
does number adding order affects to the final sum?
when the number of numbers are finite, obviously final sum will not depend on the order they are adding(the commutative law). but when the number of numbers are infinite, does the final summation ...
1
vote
1answer
92 views
Sum $\sum \frac{1}{n}\not \in N$ [duplicate]
If $S_n$ denote sum of $n$ terms of H.P. $\frac{1}{2},\frac{1}{3},\frac{1}{4}$ ..... , Then prove using summation of series that $S_n\not\in N$ $\forall \ n \in N$;
1
vote
1answer
73 views
Iteration of the function $d(n)=a-n$
I start by defining the function $f$
$$f(0)=0,~~~~~f(n+1)=d(f(n))=a-f(n)$$
So:
$$f(n)=d^{\circ n}(0)$$
$f(1)=a-0=a$
$f(2)=a-(a-0)=0$
$f(3)=a-(a-(a-0))=a$
How can I find the solution of $f(x), ...
1
vote
1answer
102 views
Multiplying two series together
How would I multiply two series together? Or also split them into two separate series? For example:
$$\sum_{y=1}^{b}\sum_{x=1}^{a}2^{(2x+3y)}$$
I tried multiplying the summation of $4^x$ with ...
8
votes
2answers
149 views
Solve a summation
Hi guys I have an exercise I don't know how to approach, would be cool if you could give me a tip or two! A sequence $a_{n}$ is defined by a dependency : $$ \sum_{i, j, k \geq 0}^{i+j+k = n } ...
3
votes
0answers
80 views
How prove this summation
prove that:
$$\dfrac{n}{n+1}+\dfrac{2n(n-1)}{(n+1)(n+2)}+\dfrac{3n(n-1)(n-2)}{(n+1)(n+2)(n+3)}+\cdots=\dfrac{n}{2}$$
I think can prove by the probability
my idea:
...
1
vote
0answers
44 views
Series sum and integral inaccuracy problem
Well, it happened that math is not my field. Having a vague concept in general I have been struggling for a few hours with the exercise. In general, I want to calculate how much interest a man have ...
5
votes
1answer
70 views
how to evaluate a sum of powers of central binomial coefficients?
I have a series I was wondering how to evaluate. It has a fun result and I am wondering how one deals with sums of ,or alternating sums of, central binomial coefficients if they're cubed.
i.e. ...
4
votes
2answers
126 views
Random sum of random variables
Say you sum i.i.d. variables $X_i$ a total of $Y$ times. If you know the distribution of random variables $Y$ and $X_i$, what is the calculation you have to do to get the distribution of the sum?
1
vote
2answers
41 views
Summations with binomial coefficients:$\sum_{k=0}^{n}\binom{R}{k}\binom{M}{n-k}=\binom{R+M}{n}$
Can someone help me solve this equation? How to prove that $$\sum_{k=0}^{n}\binom{R}{k}\binom{M}{n-k}=\binom{R+M}{n}?$$
4
votes
2answers
92 views
0
votes
1answer
127 views
Split up sum of products $\sum{a_i b_i}\approx(1/N)\sum{a_i}\sum{b_i}$ for uncorrelated summands?
As the topic says, is $\sum{a_i b_i}\approx(1/N)\sum{a_i}\sum{b_i}$ possible when $a_i$ and $b_i$ uncorrelated? I have come across something like that very recently where this has been magically done ...
2
votes
1answer
42 views
Does this summation index make any sense?
From my textbook, I have this summation:
$$
y_f(k) = \sum_{\tau = k_0}^{k-1}a^{k-1-\tau}g(\tau)
$$
So far so good. But then there is a "change of variable" $\tau = \theta - m$ and the summation ...
0
votes
1answer
33 views
Summation involving the index multiplied by an exponential?
$$\begin{align}
\sum_{i=0}^{n-1}ia^i &= 0 + a + 2a^2 + \cdots + (n-1)a^{n-1}\\
&= \frac{a - na^n + (n-1)a^{n+1}}{(1 - a)^2}
\end{align}$$
Is there any formula for a slightly similar one?
$$
...
5
votes
1answer
124 views
Sum the infinite series of $\frac{1}{r^3+1}$
Is there a definite value for the sum:
$S=\sum_{r=1}^{\infty} \frac{1}{r^3+1}$
And if so, how would I arrive at finding this sum?
I have tried reducing the above into partial fractions, however I ...
2
votes
4answers
102 views
Finding $ \sum_{n=1}^\infty {{1} \over ({2n-1})^2} $ given $ \sum_{n=1} ^ \infty {{1} \over {n^2}} = {{\pi^2}\over {6}}$
If $$ \sum_{n=1} ^ \infty {{1} \over {n^2}} = {{\pi^2}\over {6}},$$ find $$ \sum_{n=1}^\infty {{1} \over ({{2n-1}})^2}. $$
I tried an approach using partial fractions and tried to transform ...
0
votes
2answers
56 views
Geometric Series with fractions (summation)
I am having trouble trying to solve this summation (with fraction)
$$f(n) = \sum\limits_{i=1}^{n} \frac{-5}{6^i}$$
How do I solve this?
0
votes
0answers
51 views
Derivative of a sum with respect to the maximum step?
I am looking to take the derivative of $\partial _n(\sum_ {j=n/2}^{n} f(n,j))$ and I am not sure how to go about doing it. Can Anyone point Me in the direction of information about how to proceed? ...
1
vote
3answers
60 views
recurrence relation expanding $ij$
I need to solve this:
$\displaystyle\sum\limits_{i=1}^n$$\displaystyle\sum\limits_{j=1}^n $$\displaystyle\sum\limits_{k=1}^{i\cdot j} 1$
How do I expand the $i\cdot j$ part? Am I right to do it this ...
8
votes
1answer
146 views
Prove that $\sum_{k=0}^n{e^{ik^2}} = o(n^\alpha)$, $ \forall \alpha >0$
I want to prove that :
$\sum_{k=0}^n{e^{ik^2}} = o(n^\alpha)$, $ \forall \alpha >0$ when $n$ tends to $+\infty$
Perhaps $\sum_{k=0}^n{e^{ik^2}}$ is bounded, I don't know.
Do you have ideas ?
5
votes
2answers
101 views
Sum up to number $N$ using $1,2$ and $3$
So the question asked was finding out the number of ways(combinations), a given number $N$ can be formed using the sum of $1,2$ or $3$.
(eg)
For $n = 8$, the answer is $10$
The given solution for ...
11
votes
3answers
272 views
A finite summation involving $2013$
Can you help me compute the summation below?
$$1+\frac{1}{2}+\cdots+\frac{1}{2013}+\frac{1}{1\cdot2}+\frac{1}{1\cdot3}+\cdots+\frac{1}{2012\cdot 2013}+\cdots+\frac{1}{1\cdot2\cdots2013}$$
4
votes
2answers
53 views
Does the following relation holds for infinite sums?
Hey i am only wondering if it is possible that the equation below holds:
$$
\sum\limits_{i=0}^{\infty} a_i(b_i - c_i) = \sum\limits_{i=0}^{\infty} (a_ib_i - a_ic_i) =\sum\limits_{i=0}^{\infty} a_ib_i ...
1
vote
2answers
72 views
Combinatorics — Fibonacci
For the following expression, find a simple formula which only involves one Fibonacci number.
Then prove it by induction.
$$F_1+F_3+ \cdots +F_{2n+1} $$
I'm be appreciated for any help. I have no ...
2
votes
3answers
76 views
Combinatoric Explanation of General Identity
When $k \lt n$, what is the value of the sum $$\sum\limits_{j=0}^n {n \choose j}(-1)^j (n-j)^k.$$ Explain combinatorially.
Any ideas on where to start?
1
vote
0answers
55 views
Analytical solution for a variable inside of a summation
I am trying to figure out how to solve the following expression for $x$ and I'm surprised that I don't know what to do.
$$\frac{2n}{x} = \sum_{i=1}^{n} \frac{1}{x-y_{i}}$$
We have that $n$ and ...
4
votes
3answers
109 views
Solve $\sum nx^n$
I am trying to find a closed form solution for $\sum_{n\ge0} nx^n\text{, where }\lvert x \rvert<1$.
This solution makes sense to me:
$\sum_{n\ge0} x^n=(1-x)^{-1} \\
\frac{d}{d x} \sum_{n\ge0} x^n ...
2
votes
1answer
195 views
A convergence problem： splitting a double sum
I have been facing some difficulties with the following question.
For an absolutely convergent series $\sum_m a_m$, and the Mobius function $\mu(n)$, $x=(x_1,x_2)\in \mathbb{R}^2$, and $\alpha ...
1
vote
0answers
28 views
Cancellation of summations
I am working on some stuff related to the convolution property of the discrete Fourier transform. If we consider:
$$\sum_{p = 0}^{N-1}\hat{s}_{p}e^{ik_{p}x_{m}} = \sum_{p = ...
1
vote
1answer
16 views
how to get a binomial from a summation
An urn contains 6 Red balls and 1 Blue ball. A fair die having faces f1;2;3;4;5;6g is
rolled. If the top face on the die shows m, then m random balls are removed from the urn.
What is the expected ...
5
votes
5answers
250 views
Function of $ \sqrt{2+\sqrt{2+\sqrt{2+}}}\ldots $
$$\sqrt{2+\sqrt{2+\sqrt{2+}}}\ldots$$ How you could put this into a summation equation? I'm stuck. At one point thought it was this:
$$ \sum_{n=0}^{x} 2^{\frac{1}{2^x}} $$
but that would just be ...
2
votes
1answer
71 views
Having difficulty with Summation
How would I compute:
$$\sum_{n=2}^\infty \frac{1}{n^2 - n} \cdot n$$
Hints or step by step process would be the most helpful.
0
votes
0answers
30 views
summation formula involving mertens function
from the residue theorem
$$ M(x) = \sum_\rho \frac{x^\rho}{\rho \zeta'(\rho)} - 2+\sum_{n=1}^\infty \frac{ (-1)^{n-1} (2\pi )^{2n}}{(2n)! n \zeta(2n+1)x^{2n}} $$
asssuming there are no multiple ...
3
votes
0answers
85 views
How to calculate the sum of floors [duplicate]
The problem is to calculate the value
$$ \sum_{i = 1}^{1000} \left\lfloor \frac{1}{\sqrt{i} - \lfloor \sqrt{i} \rfloor} \right\rfloor $$
Where $ i $ is not perfect square number.
My thought is, if ...
3
votes
1answer
75 views
A summation of a series with power, exponential and factorial
$\sum_{k=0}^\infty {\frac{k^{C_1} {C_2}^k}{k!}}$
where $C_1$ is a positive integer and $C_2$ is a real number. Is there a close form or an approximation of the result when the summation converges?
1
vote
3answers
85 views
Boundedness of the sum of $\arctan n$
Could someone please explain me why $$s_n=\sum^n_1 \arctan(k)$$ is bounded?
P.S I need it to understand something. Thanks beforehand.
5
votes
1answer
78 views
How to compute the sum of every $k$-th binomial coefficient?
My teacher was discussing binomial expansions of $(1 + x)^n$ and he gave as an interesting example with $x = i$ whereby you could obtain the sum of all the odd coefficients ($C_n^1+ C_n^3+ C_n^5 ...$) ...
0
votes
1answer
20 views
How to handle the summation
In one of the text book I found the following expression
$\sum_{k=0}^{\infty} \sum_{s=0}^{\infty} h(k) g(s) z^{-(k+s)}$
Let $k+s=l$
Then In the book it has written the following
...
