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

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4
votes
1answer
121 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
41 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
vote
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
24 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
votes
1answer
38 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
vote
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
votes
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
31 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
vote
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
149 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
33 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 $$\sum_{n \in \mathbb Z_+} \frac{2^{n-1}}{2^{2^n}}$$
2
votes
1answer
22 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
vote
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
votes
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
votes
2answers
17 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
votes
2answers
48 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?
0
votes
0answers
40 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
votes
0answers
10 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
votes
2answers
95 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
votes
4answers
171 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
votes
0answers
65 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} ...
1
vote
2answers
42 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
votes
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
33 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: ...
0
votes
3answers
42 views

How do i evaluate a nested summation with fraction?

i have to evaluate this expression, but im not sure how to begin. $$\sum^{4}_{i=1}\sum^{5-i}_{j=2} \frac{(j+1)^2}{(2i-1)}$$
1
vote
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? $$ \sum_{k=1}^{\infty} k \cdot \frac{1}{x^k - x^{-k}} $$
1
vote
0answers
18 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
293 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
votes
0answers
39 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.
0
votes
1answer
26 views

Prove that a Sequence Approaches Infinity

I have to calculate the limit of the following: $\lim\limits_{n \to \infty} (\frac{n}{n+1}\sum_{k=0}^n\frac{k}{k+1})$ I think that the answer is infinity. Explanation: $\lim\limits_{n \to \infty} ...
0
votes
0answers
13 views

Help simplifying this sum $f(x) =\sum_{n=1}^{\infty} \frac{2x}{n} e^{-x^2/n} 2^{-n}$, $ x \ge 0$

I am stuck on this sum $f(x) = \sum_{n=1}^{\infty} \frac{2x}{n} e^{-x^2/n} 2^{-n}$ $ x \ge 0$ Any tips on how to get started? Thanks for any help
1
vote
2answers
23 views

Sum with non unit increment

Let's consider the sum $$\sum_{i=4t+2} {\binom{m}{i}}$$. It's equivalent to the following $\sum_{s}{\binom{m}{4s+2}}$, but i got stuck here. How to evaluate such kind of sums? For instance, it's ...
1
vote
0answers
8 views

Probability of summation of i.i.d. variables with a spherical joint distribution

I have a question regarding the probability of summed i.i.d. variables (log-returns) that have a joint spherical distribution. Obviously, the following statement holds: $$ P(X_1 + ... + X_{10} < ...
1
vote
1answer
19 views

How can I find the sum of any homogenous linear recurrence relation?

I've become interested in linear recurrence relations of the form $a_n=-a_{n-1}-a_{n-2}- ... $ where $a_0=1$. For the first of these relations I considered $a_n=-a_{n-1}-a_{n-2}$ where $a_0=1$ and ...