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

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1answer
49 views

Solve this combination with a summation. Edited and maybe Solved [duplicate]

I have been quite stuck on this problem and with the help of others, I may have solved it. This is what I have to prove. $$\binom{N}{n+1}=\sum^{N-1}_{k=n} \binom{k}{n}.$$ So far I have this $${N ...
0
votes
1answer
17 views

proving $(1+\frac 1n)^{n} = 1 + \sum_{k=1}^n \{\frac 1{k!}\prod_{r=0}^{k-1}(1-\frac rn)\}$ using the binomial theorem

$(1+\frac 1n)^{n} = 1 + \sum_{k=1}^n \{\frac 1{k!}\prod_{r=0}^{k-1}(1-\frac rn)\}$ this exercise is taken from Apostol's Calculus I (page 45) and it's supposed to be proved by using the binomial ...
9
votes
4answers
133 views

How to prove that $ \sum_{k=1}^{n-1} \frac{1}{1-e^{2 \pi i k/n}} = \frac{n-1}{2}$?

I came across the fact that $$ \sum_{k=1}^{n-1} \frac{1}{1-e^{2 \pi i k/n}} = \frac{n-1}{2}.$$ How can we prove this identity?
2
votes
1answer
45 views

Showing that this sum is equal to the fibonacci numbers

How do I show that the following sum is equal to the fibonacci numbers? Atleast numerical evaluation suggests it is $$ \sum_{k=0}^{\lceil n/2\rceil}\binom{n+1-k}{n+1-2k} $$ The image below shows how ...
0
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0answers
42 views

What is the sum of reciprocals of Natural Numbers? [duplicate]

I want to calculate the sum of first $n$ natural numbers. I used the following C program to compute the first '$n$' digits : ...
2
votes
3answers
74 views

Compact form of sum $\sum\limits_{k=0}^m (-1)^k \binom{n}{k} \binom{n}{m-k}$

How to find compact form of the sum $$\sum\limits_{k=0}^m (-1)^k \binom{n}{k} \binom{n}{m-k}$$ It looks like it's connected with Vandermonde's identity but I couldn't get to the solution.
0
votes
1answer
33 views

Upper bound to multinomial coefficient sum

I'm currently stuck on what seems like a very trivial problem. I have the following calculation $$ \sum_{k_1+k_2=0}^{n} {n \choose n - k_1 - k_2, k_1, k_2}^2 \le \sum_{k_1+k_2=0}^{n} {n \choose ...
4
votes
0answers
42 views

sum of center binomial coefficients over exponential

I'm trying to find a closed form for the following sum, if anyone knows a way, a hint would be much appreciated... $$ X(n) = \sum_{i=1}^n \frac{i \choose \left \lfloor{i/2}\right \rfloor }{2^i}\ $$ ...
3
votes
0answers
33 views

Partial sum of squared binomial coefficients

Is there any formula for the partial sum of squared binomial coefficients $$S_n(k):= \sum_{i=0}^{k} \binom{n}{i}^2,$$ where $k<n$? Thank you in advance.
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2answers
35 views

Find formula for nth term of series (confused how to do it).

So my question is find a formula for the sum of the first $n$ terms of the series (not sequence): $$\frac{4}{81}+\frac{4}{729}+\frac{4}{6561}+\frac{4}{59049}+\ldots~\textrm{upto }n\textrm{ terms}$$ ...
2
votes
2answers
116 views

How to calculate $\sum_{m=1}^{N}\binom{m+k-1}{m}$. [closed]

What would be a simplified formula for $\displaystyle \sum_{m=1}^{N}\binom{m+k-1}{m}$ for a given number $k$ and any number $N$?
1
vote
7answers
45 views

How to derive $\sum_{k=0}^{n}2^{k}(n-k) = 2^{n+1} - n - 2$?

In answering this question, I thought about working out a closed-form formula for $f(n)$ there. I got as far as writing: $$ f(n) = \sum_{k=0}^{n}2^{k}(n-k) $$ …but I wasn't sure how to go farther. I ...
0
votes
1answer
46 views

Equivalence of sums

I was hoping someone might be able to help me justify this sum equivalence I ran across in a proof. I'm sure it is something simple but never the less I am confused. I have the following for a ...
0
votes
1answer
75 views

Summation of binomial coefficients [duplicate]

Is there a closed formula for: $\sum_{i=1}^{N}{\binom{i+k}{i}}$ ( k is a constant whole number )
0
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2answers
68 views

Evaluating $\sum_{n=1}^{\infty}\frac{n}{(n+2)!}$

I need to evaluate $$\sum_{n=1}^{\infty}\frac{n}{(n+2)!}$$Answer in book and WolframAlpha both say that is equal $3-e$. Thus, I have mistake and got: $$ \begin{align} ...
2
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3answers
56 views

Euler sum of a divergent series

So I have a series $1+0+(-1)+0+(-1)+0+1+0+1+0+(-1)+...$ Is it correct to rearrange this as $1+0+(-1)+0+1+0+(-1)+0+1+0+(-1)+0...$ The second problem can be done as an Euler sum and the answer is ...
-1
votes
1answer
52 views

Calculating $\sum_\limits{z=2}^{\infty} \frac{(1/2)^z}{(z - 2)!}$ [closed]

What are the steps involved to calculate the following series: $$\sum_\limits{z=2}^{\infty} \frac{(1/2)^z}{(z - 2)!}\;?$$
2
votes
0answers
161 views

An improvement of Jensen's inequality - help please!

It would nice if someone could help me with this problem. I am looking at an improvement to the classical Jensen's Inequality: $$\int_\limits{}^{} \phi(x) \mu \mathrm{d}x \geq ...
3
votes
2answers
141 views

Sum of combinations with varying $n$ [duplicate]

What is the sum of number of ways of choosing $n$ elements from $(n+r)$ elements where $r$ is fixed and $n$ varies from $1$ to $m$ ? Can this be reduced to a formula ? $$ \sum ^m _{n=1} \binom{n + ...
4
votes
2answers
131 views

Is there a way to simplify a sum of cosecants?

A problem I have been working on recently results in a sum of cosecant terms. Specifically, $f(n) = \sum_{k=1}^n \csc \frac{\pi k}{2n+1}$ $g(n) = \sum_{k=1}^n [(-1)^{k+1}(\csc \frac{\pi k}{2n+1})]$ ...
5
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2answers
85 views

If $\sum_{i=1}^n a_n=0$ then you can find a “good” ordering of $a_i$.

I'm trying to prove (or disprove, but I think it's true and I'll be surprised if someone would manage to disprove it) a small theorem. Given an array of real numbers $A=[a_1,a_2,...,a_n]$ such that ...
1
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1answer
19 views

Summing up decrementing geometric series?

Is there any easy way of summing up, $c,z \in R$ $z < 1, c < z $ $ k,n\in N$: $$\large\sum_{k=0}^{\lfloor\frac{z}{c}\rfloor}\prod_{n=0}^{k}(z-nc)^n$$ I'm searching for a formula to sum up ...
1
vote
1answer
52 views

Closed form for an infinite sum over Gamma functions?

I am having quite a bit of trouble trying to find a closed form (or a really fast way to compute) for the infinite sum $$\sum_{n=1}^{\infty} a^n \dfrac{\gamma(n+1,b)}{\Gamma(n+1)\Gamma(n)}$$ where ...
0
votes
2answers
63 views

Exponential sum [duplicate]

$$ \sum_{k=1}^\infty \frac{k^2}{2^k} = 6$$ But how? What formula should we use? It is not geometric series. Thanks Ann
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3answers
67 views

Understand the steps in a Summation

I don't understand the following steps: \begin{align*} \sum_\limits{t=1}^{n-2} (-1)^{t-1} {n \choose t}a\cdot\tfrac{1}{2}\cdot(n - t)(n - t - 1) & = a\cdot\tfrac{1}{2}\cdot n(n - 1) ...
3
votes
0answers
54 views
+50

Distance and Coordinates in fractional dimensions and the creation of functions with non-integral numbers of paramters.

Background: The Euclidean distance between two points in $n$ dimensions, where $n$ is a positive integer, and position can be described by a vector is given by... $$D_E=\left(\sum_{k=1}^n ...
4
votes
1answer
119 views

Complex Analysis proof of multinomial expression

I've recently come across the following identity $$ \displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n \choose n} $$ A nice complex analysis proof (by Felix Marin, here) follows as: ...
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votes
2answers
40 views

How do I solve a summation with n as upper limit? [closed]

How do I go about computing $$\sum\limits_{i=28}^n \left(3i^2-4i+\dfrac{5}{7^i}\right)$$ ?
1
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0answers
26 views

Closed form for $\sum_{k\in\mathbb{N}}\frac{k}{a\uparrow^kb}$

Let $a,b\in\Bbb{N}$. Is there a closed form for $\displaystyle\sum_{k\in\mathbb{N}}\frac{k}{a\uparrow^kb}$ ? (I use Knuth's up arrow notation) If so, how can we obtain it ? If there isn't a closed ...
0
votes
1answer
44 views

Closed form of a sum involving powers

How can one prove this equality ? $$\sum_{k=m}^\infty \frac{(mp)^k}{m!m^{k-m}}\quad =\quad \left(\frac{(mp)^m}{m!}\right)\left(\frac 1{1-p}\right), \quad p\lt 1$$
3
votes
1answer
23 views

Convergent series? Gamma/power function

Is it true to use as a general rule of thumb that the Gamma function always "kills" power function in a series? I mean: $$\sum_{n=1}^{\infty} \frac{C^n}{\Gamma(n)^p}<\infty$$ no matter the constant ...
0
votes
1answer
33 views

$λ={41/10\left(\frac{1}{2^2-1}+\frac{1}{4^2-1}+\frac{1}{6^2-1}+..+\frac{1}{40^2-1}\right)}$ then $w+w^λ$ is equal to

Given that, $$λ={41/10\left(\frac{1}{2^2-1}+\frac{1}{4^2-1}+\frac{1}{6^2-1}+..+\frac{1}{40^2-1}\right)}$$ then $w+w^λ$ is equal to ? [$w$ is cube root of unity other than 1] I cannot understand how ...
-2
votes
1answer
68 views

Find $\frac{1}{3}+\frac{1\cdot 3}{3\cdot 6}+\frac{1\cdot 3\cdot 5}{3\cdot 6\cdot 9}+\cdots$ [duplicate]

$$x=\frac{1}{3}+\frac{1\cdot 3}{3\cdot 6}+\frac{1\cdot 3\cdot 5}{3\cdot 6\cdot 9}+\cdots$$ How to sum this? I see that the numerator and denominator are different APs.
0
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1answer
37 views

Summation with factorial

I want to understand how this step is performed. Can you tell me that how this value of Po is obtained from the first equation.! ...
1
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2answers
51 views

A basic combinatorial sum

I am interested in the following, which I think is basic which I don't know how to find an upper bound for: $$ \sum_{j=1}^{d-1} \ \sum_{1 \leq i_1 \leq i_2 \leq ... \leq i_j \leq K} 1. $$ I would ...
1
vote
3answers
47 views

Error in approximating the sum

I am watching one of the online probability courses and in one of the lectures, the professor simplifies the sum: $$A = \sum_{j=0}^{N}\frac{j^k}{N^k} \cdot \frac{1}{N+1}$$ in the following way: $A ...
1
vote
2answers
54 views

Sum of increasing integer numbers

Please help me to calculate this sum: $$ \sum\limits_{1\leq i_1 < i_2 <\ldots i_k \leq n} (i_1+i_2+\ldots+i_k). $$ Here $n$ and $k$ are positive integer numbers, and all the numbers $i_1, i_2, ...
1
vote
1answer
39 views

binomial identity with negatives

Prove that $$\sum_{k=0}^n(-1)^k\binom{n+1}{k+1}(k+1)^n=0\;.$$ I tried finding a combinatorial interpretation but to no avail. Here is a combinatorial statement, however crappy. Suppose we have $n$ ...
5
votes
4answers
64 views

$\sum_{k=1}^n \log k \ge \int_1^n \log x \, dx$

Why is $$\sum_{k=1}^n \log k \ge \int_1^n \log x \, dx$$ is there an intuitive or graphical way to think about it?
0
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1answer
24 views

Summand Evaluation Help

I'm a student currently in an algorithms and data structures class, and my Calculus is unfortunately quite shaky when it comes to summations. As such, I'm struggling to evaluate one of the sum that ...
-5
votes
1answer
62 views

Estimating partial sums $\sum_{n = 1}^m \frac{1}{\sqrt{n}}$

Apostol's Calculus, exercise number I 4.7 13. Prove that if $n \geq 1$, then $$ 2(\sqrt{n+1} - \sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n} - \sqrt{n-1}) $$ and use this to prove that if ...
0
votes
1answer
47 views

Summation Sequence Question

I need to find the summation of $ab^{-k}$ from $k=5$ to $n$ using Gauss' Law. Here's what I have so far: $$\begin{align}S_n&=(ab^{-5}+ab^{-6}+ab^{-7}+\cdots+ab^{-n}+ab^{-n}+ab^{-(n-1) ...
1
vote
1answer
30 views

Summation Sequence

I'm supposed to use Gauss' law to find the summation of $6k$ from $k=5$ to $n$. Here is my work: $$6(5)+6(6)+6(7)+⋯+6(n)\\+6(n)+6(n-1)+6(n-2)+...+6(5)$$ When these are added together I get ...
1
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4answers
47 views

$\sum _{n=1}^{\infty }u^{n-1}=\frac{1}{1-u}$ is available for all cases?

I don't understand this equal $\sum _{n=1}^{\infty }u^{n-1}=\frac{1}{1-u}$ . I verify all value and don't get $\frac{1}{1-u}$ Please explain why we obtain that equal. I obtain $\frac{1-u^n}{1-u}$ ...
1
vote
3answers
148 views

How we get to find the result of this limit?

$$\lim _{n\to \infty }\left(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots+\left(-1\right)^{n-1}\cdot \frac{1}{2n-1}\right)=\text{ ?}$$ I don't know how we get to find the result of this operation...
0
votes
0answers
89 views

Normally Distributed Summation of Random Variable

Suppose that at ABC Company there is only one customer representative. Let N Bin(10, 0.6) be the number of customers requiring service in one hour, and Si N(10, 5) be the service time (in minutes) ...
0
votes
1answer
59 views

Operation with Sigma

How demonstrate that operation: $$1)\sum _{k=1}^{2n}\frac{\left(-1\right)^{k-1}}{k}\:+2\sum _{k=1}^n\left(\frac{1}{2k}\right)\:=\:\sum _{k=1}^{2n}\left(\frac{1}{k}\right)$$ $$2)\sum ...
-1
votes
3answers
31 views

Summation and sequence series question

Sum of fifty positive nos. is 1. Find maximum value of sum of their inverse. I have no idea how to solve this question... do not mark it as off topic or anything... Maybe we should use AM>=GM?
0
votes
1answer
24 views

Why does the unit vector of form $x_i=\frac{-1}{\sqrt{n}}$ minimize sum of $x_i$?

Cauchy-Schwarz implies that for $||\vec{x}||=1, \vec{y}=(1,\ldots,1)\in\mathbb R^n,\sum_{i=1}^{n} x_i = \pm\sqrt{n}$ if $\vec{x}=\pm{k}\vec{y}$. This implies that ...
2
votes
0answers
71 views

How to find $\sum_{n \in \mathbb Z_+} \frac{2^{n-1}}{2^{2^n}}$?

I'm trying to calculte the measure of a fat Cantor set, but run into this question: How to find $$\sum_{n \in \mathbb Z_+} \frac{2^{n-1}}{2^{2^n}}$$