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

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2answers
27 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 ...
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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) ...
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2answers
19 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 ...
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2answers
60 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
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 ...
9
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2answers
127 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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4answers
194 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
66 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
44 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 ...
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1answer
30 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
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2answers
49 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: ...
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3answers
45 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 $\lim_{x\rightarrow\infty}A$ ...
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0answers
45 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
31 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 ...
9
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4answers
334 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
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1answer
35 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 $i< j$ ...
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1answer
29 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} ...
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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
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2answers
27 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 ...
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0answers
10 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} < ...
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1answer
20 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 ...
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1answer
14 views

How can I re-write $H(X_1,X_2)+H(X_2,X_3)+H(X_1,X_3)$ using $\sum$ notation?

How can I re-write $H(X_1,X_2)+H(X_2,X_3)+H(X_1,X_3)$ using $\sum$ notation? Also how can I re-write $H(X_1,X_2,X_3)+H(X_1,X_2,X_4)+H(X_1,X_3,X_4)+ H(X_2,X_3,X_4)$ using $\sum$ notation? Is there ...
0
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0answers
27 views

Binominial Theorem proving

As I was trying to understand the proof of Binomial Theorem by induction, I got stuck at this line. What formulas should be used to get from left to right part? Any explanations and answers ...
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99 views

How to prove $\sum_{k=0}^n \binom{n}{k}=2^n$ and $\sum_{i=1}^n i(n-i+1)= \binom{n+2}{3}$ by induction? [closed]

Prove by induction that: $\sum_{k=0}^n \displaystyle\binom{n}{k}=2^n$. Hint: When you consider this equality for $n-1$, add it to itself and use a famous property of the binomial ...
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1answer
26 views

Prove absolute convergence for a summation

I need help with this problem. I've been staring at the page blankly tyring to think of ways to solve it. Any hints/solutions would be greatly appreciated. If $ \displaystyle \lim_{n \to ...
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1answer
16 views

Is it possible to find $n$ from the sum forming a polynomial?

How does one solve for $n$ in: $100000 = \sum\limits_{x=1}^n 1020.2065\ x^{-0.3431}$
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0answers
26 views

Orthogonal spaces, span-formula

I read in a book this formula: ($v_i$ are vectors of an euclidean vector space, each one $\neq$ 0) $(\cap v_i ^\bot )^\bot = \sum v_i^{\bot \bot}$, The intersection and the sum are build over a ...
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0answers
46 views

Summation involving digamma and floor functions

I am trying to find an asymptotic expansion for the following sum: $$\sum_{n=1}^K \frac{\phi_0( 1/2+n+\lfloor(2n-1)/\sqrt{2}\rfloor)}{(4n-2)}$$ where $\phi_0$ is the digamma function and $\lfloor ...
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1answer
19 views

some questions about interchange summation signs of multi-summation

Besides e.g. $\sum\limits_{b=c}^d\sum\limits_{a=c}^bf(a,b)=\sum\limits_{a=c}^d\sum\limits_{b=a}^df(a,b)$ , are there any further good formulae about interchange summation signs of multi-summation? ...
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1answer
113 views

Expectation of minimum and maximum of sum of iid random variables?

Looking for $\mathrm{E}[\min(\sum{X}) ]$ and $\mathrm{E}[\max(\sum{X})]$. Paper references much appreciated. Model: let's say we have 3 connected devices in a signal processing pipeline: $$ ...
2
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4answers
183 views

Find the sum of the series $\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}-\cdots$ [duplicate]

My book directly writes- $$\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}-\cdots=-\ln 2+1.$$ How do we prove this simply.. I am a high school student.
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0answers
25 views

Checking the correctness of my triple-nested loop analysis

I am trying to analyze the complexity of a triple-nested loop. Using previous posts here as a guidance, I believe I have arrived at the correct solution. However I would appreciate if somebody could ...
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2answers
82 views

Why Does $ \sum\limits_{k=0}^n \begin{pmatrix} n+1 \\ k+1 \end{pmatrix} p^{k+1} (1-p)^{n-k} $ sum to $ (1-(1-p)^{n+1}) $?

I was browsing around when I found this question: Find the expected value of $\frac{1}{X+1}$ where $X$ is binomial. I understood the solution until I hit this portion where $ \sum\limits_{k=0}^n ...
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1answer
55 views

Probability that two numbers in different numeral system match a pattern

How likely is it that the following holds true. $ s_1=\sum_{i=0}^{9}{16^{i} \cdot x_i}$ where $ x_i\in[0,15]\cap\mathbb{N}$ and $x_i=0$ $\forall$ $i \in\{7,6,3,2\}$ $ s_2=\sum_{i=0}^{7}{36^{i} ...
9
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1answer
190 views

Show that $\lim_{n\to\infty}(6n)^{\frac16}a_n=1$ with $(a_n)$ such that $\lim_{n\to\infty}a_n\sum_{j=1}^na_j^5=1$

Show that $$ \lim_{n\to\infty}(6n)^{\frac16}a_n=1, $$ where $(a_n)$ is a sequence of nonnegative real numbers such that $\lim_{n\to\infty}a_n\sum_{j=1}^na_j^5=1.$ I recently got stuck on this ...
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1answer
47 views

Summation and minimal value function

I am working on a summation problem that is asking me to find the sum of an expression with the minimum value function in the exponent. I'm not sure about the rules when working with sums and ...
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2answers
41 views

Sum of arithmetic progression

While solving discrete math problem I've got the sequence of positive whole numbers defined like this (I've looked up simplification to arithmetic progression in the answers): $$ (n-2) + (n-3) + ... = ...
5
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1answer
150 views

Prove this Floor function indentity $\sum_{k=0}^{n-1} \bigl\lfloor \frac{ak+b}{c} \bigr\rfloor$

Assume $a,b,c$ be positive integers. Show that: $$\sum_{k=0}^{n-1} \left\lfloor \frac{ak+b}{c} \right\rfloor = \sum_{k=0}^{\left\lfloor \dfrac{an+b}{c}\right\rfloor} ...
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0answers
40 views

$\left(\sum_{i=1}^n1\right)\left(\sum_{i=1}^n x_i^2\right)-\left(\sum_{i=1}^n x_i\right)^2=\frac{1}{2}\sum_{i=1}^n\sum_{k=1}^n\left(x_i-x_j\right)^2$?

I am reading Widder's Advanced Calculus and on page 130 he states that \begin{align}\left(\sum_{i=1}^n 1\right)\left(\sum_{i=1}^n x_i^2\right)-\left(\sum_{i=1}^n ...
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3answers
50 views

Special Case of Summation

Hello what would be the solution to the summation over the range from 1 to 0? $$ \sum_{1}^{0} = ? $$ My guess is -1 or 0, but I can't find any reference to this case.
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0answers
33 views

Find general expression for sum of sequence

How would one go about to prove that these expressions are equal? $$ \sum_{k=0}^{min(X,34)} \frac{1}{1.03^{k}}\ = \frac{1-(1/1.03)^{min(X,34)+1}}{1-1/1.03} $$ Wolfram alpha gave me an answer to the ...
0
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1answer
43 views

Sum of power functions over a simplex

Let $d \ge 1$ be a positive integer. Let $n,m$ be another positive integers subject to $m\ge n+d$. Let $\vec{x} := (x_1,x_2,\cdots,x_d)$ be real numbers such that all of them cannot be equal to ...
1
vote
1answer
52 views

Sum of Some Binomial Terms Equals Zero

Let $q$ and $\ell$ be positive integers. Then the sum $$ \sum_{k=q}^\ell (-1)^{k+q}\binom{k}{q}\binom{\ell}{k} = \left\{\begin{array}{ccc} 1 \mbox{ if } \ell =q\\ 0 \mbox{ if }\ell ...
5
votes
3answers
121 views

How to evaluate $\sum_{n=2}^\infty\frac{(-1)^n}{n^2-n}$

How would you go about evaluating:$$\sum_{n=2}^\infty\frac{(-1)^n}{n^2-n}$$ I split it up to $$\sum_{n=2}^\infty\left[(-1)^n\left(\frac{1}{n-1}-\frac{1}{n}\right)\right]$$ but I'm not sure what to ...
1
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0answers
161 views

sum of exponential series with power increasing by geometric series

Is there any way to reduce the following summation... $2^{ar^0}+2^{ar^1}....+ 2^{ar^n} $ to a simple equation? I feel like I can pull out a $2^a$ somehow and then treat it as a normal series but I ...
0
votes
0answers
33 views

derivative on sum operator

I have the following formula: $$ \frac{d}{da[k]}E[e[n]^2] $$ Where $$ e[n]=\sum^{p}_{k=1}a[k](x[n-k]-\hat{x}[n-k]) $$ is the result: $$ ...
3
votes
1answer
38 views

Expressing $\sum_{k=1}^{n}\frac{1}{(k+2)k!}$ in terms of $n$.

How would I express $$\sum_{k=1}^{n}\frac{1}{(k+2)k!}$$ in terms of $n$? An attempt of mine is $$\sum_{k=1}^{n}\frac{1}{(k+2)k!} = \sum_{k=1}^{n}\frac{1}{(k+1)! + k!},$$ which is not useful for ...
1
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2answers
37 views

How does $\sum (Y_i-\bar{Y})^2 = \sum Y_i^2 - n\bar{Y}^2$?

I've tried my algebra backwards and forwards and starting from the left-hand side of the equation below I just can't get to the right-hand side. I'm always left with an extra term $-2Y_i\bar{Y}$. ...