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

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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
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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 ...
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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 $i< j$ ...
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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
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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} ...
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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
1
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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 ...
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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
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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 ...
0
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1answer
13 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
24 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 ...
1
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0answers
79 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 ...
1
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1answer
24 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 ...
0
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1answer
11 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
24 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
32 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
16 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? ...
-1
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1answer
66 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
158 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
18 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
74 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
48 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
186 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 ...
0
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1answer
30 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 ...
1
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2answers
27 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) + ... = ...
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0answers
39 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
46 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
30 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
32 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
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1answer
49 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
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3answers
114 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
48 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
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0answers
13 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
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1answer
37 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
32 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 $-2Y_i\bar{Y}$. ...
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1answer
40 views

prove that if m and n are any positive integers and m is odd, then $ m \mid \sum \limits_{i=0}^{m-1} (n ~+~ i)$ is divisible by m.

Trying to figure out the induction prof on this theorem: $$ \forall m,n \in \mathbb{Z}, ~ m,n \geq 1 ~\land~ m \equiv 1(\mod 2) ~\rightarrow~ m \mid \sum \limits_{i=0}^{m-1} (n ~+~ i) $$ I got the ...
4
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1answer
74 views

Sum and Product Puzzle and Prime Factors

Suppose we have two number $X$ and $Y,$ such that $1 < X < Y < 100,$ and $X + Y ≤ 100.$ Sue is given $S = X + Y$ and Pete is given $P = XY.$ They then have the following conversation: ...
2
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2answers
60 views

Summation over a product of binomial coefficients

Question: I can't figure out why the following equality is true $\sum_\limits{k=a-b-c}^{d} (-1)^k \binom{d}{k}\binom{k+b+c}{a} = (-1)^d \binom{b+c}{a-d} $ How can this be shown? (In the book it just ...
0
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1answer
26 views

Summation problem (probability)

I have the equation $$\Pr(X\le6)=\sum_{x=6}^{∞}\left({e^{-4.8}}\cdot\frac{4.8^{x}}{x!}\right).$$ And it is not equating to when I sum each term manually. Plugging this into my calculator I get ...
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3answers
72 views

Combinatorial proof of $ \sum \limits_{i = 0} ^{m} 2^{n-i} {n \choose i}{m \choose i} = \sum\limits_{i=0}^m {n + m - i \choose m} {n \choose i} $

I've been wondering for a while how to solve (prove) a combinatorial identity, using just combinatorial interpretation: $$ \sum \limits_{i = 0} ^{m} 2^{n-i} {n \choose i}{m \choose i} = ...
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4answers
796 views

Where do summation formulas come from?

It's a classic problem in an introductory proof course to prove that $\sum_{ i \mathop =1}^ni = \frac{n(n+1)}{2}$ by induction. The problem with induction is that you can't prove what the sum is ...
4
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3answers
55 views

Evaluating $\sum_{0\leq k,l \leq n}\binom{n}{k}\binom{k}{l}l(k-l)(n-k)$ algebraically

I'm having problems with the following sum: $$\sum_{0\leq k,l \leq n}\binom{n}{k}\binom{k}{l}l(k-l)(n-k)$$ It's quite easy to think about it combinatorically: We have $n$ balls, we're coloring $k$ ...
2
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2answers
49 views

Computing $\sum_{i=2}^{n-1} 1$ and $\sum_{i=2}^{n-1} i$.

Could someone explain how I calculate these summations? I'm using upper - lower + 1. a) $\displaystyle\sum_{i=2}^{n-1} 1$ b) $\displaystyle\sum_{i=2}^{n-1} i$ So for (a) I have: $$ (n-1) -2 +1)1 ...
3
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2answers
199 views

A “generalized” exponential power series

I'm wondering if $$ e^x = \sum_{k=0}^\infty \frac{x^k}{k!} $$ what would this be $$ \sum_{k=0}^\infty \frac{x^{k+\alpha}}{\Gamma(k+\alpha)} = \large{?}_{\alpha}(x) $$ for $\alpha \in (0,1)$? ...
1
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5answers
44 views

Sumfunction of $\sum_{n=0}^{\infty} \frac{z^{4n}}{(4n)!}$ and $\sum_{n=1}^{\infty} \frac{z^{-n}}{4n}$.

Sumfunction of $\sum_{n=0}^{\infty} \frac{z^{4n}}{4n!}$ and $\sum_{n=1}^{\infty} \frac{z^{-n}}{4n}$. The first one looks like the cosine hyperbolicus but the $4n!$ anoys me. I tried using ...
6
votes
1answer
51 views

Tan inverse summation

$$S=\sum\limits_{i=1}^{4}\tan^{-1} x_i$$ How to simplify this ? I think I will have to use this : but it looks too long a method . Is there a method or symmetrical way which yields ...
6
votes
0answers
95 views

Why use Einstein Summation Notation?

Einstein summation convention dictates that repeated indices should be summed. Thus the equation $a_{ij} = b_{ik}c_{kj}$ is taken to mean $a_{ij} = \sum_k b_{ik}c_{kj}$ where in both cases the range ...
6
votes
2answers
95 views

Sum of factorial fractions

Find the sum $$\sum\limits_{a=0}^{\infty}\sum\limits_{b=0}^{\infty}\sum\limits_{c=0}^{\infty}\frac{1}{(a+b+c)!}$$ I tried making something like a geometric series but couldn't. Then I couldn't think ...
5
votes
2answers
142 views

Limit of an arctangent summation

Evaluate: $$S=\tan\left(\sum_{n=1}^{\infty}\arctan\left(\frac{16\sqrt{2557}n}{n^4+40916}\right)\right)$$ For $x$ small enough, $\arctan(x)<x$, and by the comparison test we know that the ...