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

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1answer
29 views

Exponential sum

1/2+4/4+9/8+16/16+25/32+36/64+⋯= 6 But how? What formula should we use? It is not geometric series. Thanks Ann
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2answers
46 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) ...
1
vote
0answers
19 views

Distance and Coordinates in fractional dimensions

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 ...
2
votes
0answers
41 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: ...
-2
votes
2answers
34 views

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

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
23 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
39 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
19 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
30 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 ...
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votes
1answer
64 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
35 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
45 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
40 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
52 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
34 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
63 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?
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1answer
22 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 ...
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votes
1answer
53 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
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1answer
44 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
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1answer
27 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
44 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
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3answers
142 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
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1answer
58 views

Demonstrate $\sum _{k=1}^n\frac{1}{2n-k}$ [on hold]

Starting from $\sum _{k=1}^n\frac{1}{k\cdot 2^k}$, find $\sum _{k=1}^n\frac{1}{2n-k}$
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0answers
69 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
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2answers
52 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 ...
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votes
3answers
28 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
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1answer
23 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
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0answers
66 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 $$\sum_{n \in \mathbb Z_+} \frac{2^{n-1}}{2^{2^n}}$$
1
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1answer
17 views

Show $\sum_{n\leq k\leq 2n}2^{-2k}\log(k)\leq C\, 2^{-2n}\log(n)$

I'd like to prove $$\sum_{n\leq k\leq 2n}2^{-2k}\log(k)\leq C \, 2^{-2n}\log(n),$$ where $C>0$ is a constant. Can someone give me a hint.
1
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2answers
22 views

Proving associativity of product of two formal sums $\sum_{n = 1}^{\infty} \frac{a_n}{n^x}$

Let $R$ be the set of all formal sums $\sum_{n = 1}^{\infty} \frac{a_n}{n^x}$ where $a_n \in \Bbb{Q}$, where two sums $a, b$ are equal iff $a_i = b_i \ \forall i$. It is indeed a ring with addition ...
10
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4answers
1k views

If the earth's rotational speed increased by 2% each day starting today…what would be the difference in age 20 years from now?

If the new adjusted revolution of the earth still equaled one day and 365 days still equaled one year, how old would someone be 20 years from now (20 years based on the current rotation of the earth) ...
2
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2answers
16 views

general sum notation considering also not incremental indexing

I need to write a formula with summation in a general case allowing also the case with not incremental indexing. Example: $ \sum_{i=\underline{i}}^\bar{i}$ where can be ...
0
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2answers
45 views

Double summation switch if one index is infinite and the other finite?

Is the following equation generally true? $$\sum_{i=1}^n \sum_{j=1}^\infty\left(a_{i,j}\right)=\sum_{j=1}^\infty \sum_{i=1}^n\left(a_{i,j}\right)$$ If true, how would you prove it?
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0answers
37 views

Variance of a Sum (Normally distributed)

How do you find the variance of Z(t) ? I've tried the following but unsure how to continue, or if this is even correct. Please help. Var(Z(t)) = Var(sum of Xi +Y) ...
0
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0answers
9 views

Upper bound $f(t,d)$ such that $\sum_{j=1}^{d} cos(2 \pi j \; t) \leq f(t,d)$. [duplicate]

I have a sum of a series of trig functions as follows: $\sum_{j=1}^{d} cos(2 \pi j \; t)$ where t is just a constant. Here, we can assume $t$ is a small number and $t \neq 0$. what is the upper ...
8
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2answers
92 views

How to evaluate $ \sum\limits_{n=1}^{\infty} \left( \frac{H_{n}}{(n+1)^2.2^n} \right)$

Evaluate $$ \sum_{n=1}^{\infty} \left( \dfrac{H_{n}}{(n+1)^2.2^n} \right)$$ Where $H_{n}$ is the $n^{th}$ Harmonic Number, i.e., $H_{n} = \displaystyle \sum _{k=1}^n \frac{1}{k}$ I ...
8
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5answers
148 views

Evaluating $ \sum_{n=1}^\infty \frac{1}{n^2 2^n} $

Evaluate $$ \sum_{n=1}^\infty \dfrac{1}{n^2 2^n}. $$ I have tried using the Maclaurin series of $2^{-n}$ but it further complicated the question. Moreover, I have also tried taking help ...
2
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0answers
63 views

Bizarre differential identity.

Let $d\ge 1$ be an integer. Let $m$ and $n$ be integers subject to $m \ge n+d-1$. The question is to prove the following identity. \begin{equation} \sum\limits_{j=-1}^{d-1} ...
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2answers
36 views

Upper bound $f(t,d)$ such that $\sum_{j=1}^{d} cos(2 \pi j \; t) \leq f(t,d)$?

I have a sum of a series of trig function as follows: $\sum_{j=1}^{d} cos(2 \pi j \; t)$ where t is just a constant. I am looking for the upper bound $f(t,d)$ such that $\sum_{j=1}^{d} cos(2 \pi j ...
0
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1answer
27 views

Is there formula for $\sum_{n=-\infty}^{\infty} sinc((t-nT)/T)$ if $t$ and $T$ is known?

Is there any simple formula for $\sum_{n=-\infty}^{\infty} sinc(\frac{t-nT}{T})$, if $t$ and $T$ are given?
4
votes
2answers
29 views

Summation Proof - Permutation of indices

How do i mathematically prove that $\sum\limits_{n=1}^N b_{n+1} = \sum\limits_{n=2}^{N+1} b_n$ This was taken from the proof of telescoping Series See: ...
1
vote
1answer
67 views

Why should we care about double and iterated summations? [closed]

I am currently taking a real analysis course and have to make a presentation on double and iterated summations. As I was making the PowerPoint, I began to wonder the following: Why are double and ...
0
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3answers
41 views

How do i evaluate a nested summation with fraction?

i have to evaluate this expression, but im not sure how to begin. $$\sum^{4}_{i=1}\sum^{5-i}_{j=2} \frac{(j+1)^2}{(2i-1)}$$
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2answers
30 views

Does $\sum_{n=1}^{x-1}\frac{1}{x-n}$ has a limit as $x \rightarrow \infty$?

Consider the sum $A = \frac{1}{x-1} + \frac{1}{x-2} + \ldots + 1 = \sum_{n=1}^{x-1}\frac{1}{x-n},\quad x > 2$ Can anyone provide some hints on how to proof that the $\lim_{x\rightarrow\infty}A$ ...
0
votes
0answers
44 views

Nicer analytical expression for infinite sum

Is it possible to rewrite the following sum as a function of $x$ in a "nicer" form, where no sum appears? $$ \sum_{k=1}^{\infty} k \cdot \frac{1}{x^k - x^{-k}} $$
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0answers
17 views

Growth of exponential sum

i am calculating large data sets with program i wrote and i have two different methods to do this. The first way is to calculate it all at once and the second way to calculate result is to do it in ...
8
votes
4answers
252 views

Orthogonality for Binomial Coefficients

Could somebody explain to me where these two formulas come from as applications of the binomial theorem? $$\sum_{k=0}^n {n \choose k}(-1)^kk^r=0$$ for non-negative integers $r\lt n$. And ...
0
votes
1answer
32 views

how to get the second equation (related to summation)

$$V(Y) = \sum_{i=1}^N\sum_{j=1}^N [\frac{N^2}{n^2}] (Y_i-Y_j)^2 \frac{n(N-n)}{N(N-1)} $$ for $i< j$ Equation(2.5) $$=(\frac{(N-n)}{n(N-1)})\sum_{i=1}^N \sum_{j=1}^N (Y_i-Y_j)^2 $$ for $i< j$ ...
0
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0answers
38 views

evaluate $\sum_{n=1}^\infty {1 \over n^2}$ [duplicate]

I'm just not sure how to go about doing sums like this, so some help evaluating the above expression please.