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

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20 views

sum of fractional order Legendre functions

I need to calculate the sum \begin{equation} \sum_{l=0}^{\infty}{\Gamma(l+N) \over \Gamma(l+1)}{\left[\Gamma\left({l \over N}+1\right)\right]^2 \over \Gamma\left(2{l \over N}+1\right)}P_{l\over N}(z) ...
0
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0answers
42 views

How is $\sum_{i=0}^{\log n -1} 2^i(\log n -i) =\sum_{i=1}^{\log n} i\frac n {2^i} $

I found the following in a textbook: $$\sum_{i=0}^{\log n -1} 2^i(\log n -i) =\sum_{i=1}^{\log n} i\frac n {2^i} $$ It's a summation of a chart, the explanation for this was to "flip the chart", I ...
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0answers
24 views

Proof of sine integral

How to prove: $$\operatorname{Si}(x) =\sum_{k=0}^\infty { \frac { -\sin\left( \frac{\pi k}{2} \right) }{ k\times k! } {(-x)}^k} $$ Here $\operatorname{Si}(x)$ is Sine Integral. Actually I ...
1
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1answer
48 views

Order of an infinite sum

How to prove that, for $0<c<1$, $$ \sum\limits_{j=1}^{\infty} c^{\, j } \cdot j^{\, -(\frac{d}{2} +1 )} $$ is, for some positive constant $K$, of order $K + O( \, ( 1-c )^{\frac{d-2}{2}})$ when ...
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3answers
56 views

prove that $ \binom{n-1}{0} +\binom{n}{1}+\binom{n+1}{2}+\cdots+\binom{n+k}{k+1}=\binom{n+k+1}{k+1}$

I am asked to prove that $$ \dbinom{n-1}{0} +\dbinom{n}{1}+\dbinom{n+1}{2}+\cdots+\dbinom{n+k}{k+1}=\dbinom{n+k+1}{k+1}$$ So far what I've tried ,without looking to much at the sum I've to prove ,is ...
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1answer
45 views

Can't understand this step in Zeta-function transformations

Sorry for exceptional dullness, but I can't understand this step in Zeta-function transformations: $$ζ(s)= \sum_{n=1}^\infty \frac{1}{n^s}= \sum_{n=1}^\infty n (\frac{1}{n^s}-\frac{1}{(n+1)^s})$$
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1answer
50 views

Calculate $\sum_{k=0}^\frac{n}{2}(-1)^k{n \choose 2k}$ and $\sum_{k=0}^\frac{n-1}{2}(-1)^k{n \choose 2k+1}$

a) $\sum_{k=0}^\frac{n}{2}(-1)^k{n \choose 2k}$ b) $\sum_{k=0}^\frac{n-1}{2}(-1)^k{n \choose 2k+1}$ I think a way to calculate the sums is to see what happens to $(1+i)^n$ but after trying for 2 ...
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1answer
34 views

Convergence of infinite sum including cosh functions?

I am attempting to code up an equation that includes an infinite sum of cosine and hyperbolic cosine functions, namely: $$ \sum_{m=0}^{\infty} \frac{ \cos[(2m+1)\pi x/s] \cosh[(2m+1)\pi z/s] } ...
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3answers
90 views

mysterious sum of two sequences

Let $$S_1 = \sum_{n=1}^\infty \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \cdots$$ $$S_2 = \sum_{n=1}^\infty ...
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2answers
42 views

Evaluate the sum $\sum_{k=0}^n 2^k \cdot \dbinom{n}{k}$

Evaluate the sum $\sum_{k=0}^n 2^k \cdot \dbinom{n}{k}$ I think the problem calls for some application of Vandermonde's identity as the author previously proved this identity,however I can't see ...
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0answers
53 views

Lost proof of trigonometric formula

The following formula seems to be true for odd positive integers $n$ but i forgot the way I proved it $$\sum_{k=1}^n\tan(\alpha+\frac{k2\pi}{n})=n\tan(n\alpha)$$ Maybe someone can deliver the ...
2
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2answers
72 views

Closed form or approximation of $\sum\limits_{i=0}^{n-1}\sum\limits_{j=i + 1}^{n-1} \frac{i + j + 2}{(i + 1)(j+1)} (i + 2x)(j +2x)$

During the solution of my programming problem I ended up with the following double sum: $$\sum_{i=0}^{n-1}\sum_{j=i + 1}^{n-1} \frac{i + j + 2}{(i + 1)(j+1)}\cdot (i + 2x)(j +2x)$$ where $x$ is some ...
-1
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3answers
89 views

Proving that $\sum_{j=0}^n(-1)^j\binom{n}{j} = \binom{n}{0} - \binom{n}{1} + … \pm \binom{n}{n}=0$ [duplicate]

The equation to be proved is: $\sum_{j=0}^n(-1)^j\dbinom{n}{j} = \dbinom{n}{0} - \dbinom{n}{1} + ... \pm \dbinom{n}{n}=0$ But if i take the base case ($n = 1$) i get $\sum_{j=0}^n(-1)^j\dbinom{n}{j} ...
4
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3answers
108 views

Induction on inequalities (a sum less than a particular value) [duplicate]

I am trying to solve this inequality by induction. I just started to learn induction this week and all the inequalities we had been solved were like an equation less than another equation (e.g. $n! ...
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1answer
61 views

The name of the sum $\sum_{i=0}^n \frac{1}{m-i}$

Sorry for the vague question name, since I am looking for the name of the series. Also this might not be a "series" by the strict definition of a series.. anyways here it is: Choose some $m$ and $n$ ...
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2answers
56 views

Induction proof without summation

I have to prove this induction: $\dfrac{1}{(n+1)}+\dfrac{1}{(n+2)}+\dots+\dfrac{1}{2n} = \dfrac{1}{(1\times2)}+\dfrac{1}{(3\times4)}+\dots+\dfrac{1}{(2n-1)\times2n}$ Can someone help me with it?
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0answers
34 views

come up with a formula for $f(n)=\sum_{i=1}^n i^k$ [duplicate]

I know that the general method is to attempt getting the constant out of the summation, then use the summation formula on that and multiply the two. Is there any way to get K outside of the summation ...
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3answers
75 views

On the sum of the reciprocals of square roots. [closed]

What is the analytic sum of $1+ \frac{1}{\sqrt 2} + \cdots + \frac{1}{\sqrt x}$ ?
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1answer
40 views

is there a difference between sum and integral?

is there a difference between integrating a function between two limits and summing a function and if so where does the difference come from and when would you use each method in real life situations
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0answers
43 views

Simplify sum with binomials

An algorithm finds prefixes of given length k from given word with length n. It is required to find the time complexity of given algorithm. It is easy when no nodes get cut off in its recursion tree ...
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0answers
35 views

How to calculate this infinite sum [duplicate]

I don't understand how to get from the infinite sum $\sum_\limits{n=1}^{\infty}\frac{1}{n^2}$ to the solution $\frac{\pi ^2}{6} $. Is it possible to calculate this by hand or do I just have to type ...
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3answers
97 views

Proof that $\sum_{i=0}^n {n\choose i}2^{i}=3^{n}$ [duplicate]

The following sum came up in a combinatorial argument. I know what it equals thanks to Wolfram Alpha, but I'm not sure how to show it $$\sum_{i=0}^n {n\choose i}2^{i}=3^{n}$$
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2answers
60 views

Writing sigma notation $\sum^n_{i=1} \frac {i}{2^i}$ in closed form

What would be a way to find the closed form of $\frac {1}{2} + \frac {2}{4}+\frac {3}{8}+\cdots+\frac {n}{2^n}=\sum^n_{i=1} \frac {i}{2^i}=s$ I've looked at $\frac {s}{2}=\frac {1}{4} + \frac ...
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2answers
29 views

Sum of first $n$ positive integers to a positive power $p$

Consider the sum $$\sum_{i=1}^{n}i^{p}\text{ , }p \in \mathbb{Z}^+\text{.}$$ Using a method in Spivak's Calculus, it can be shown that $$(n+1)^{p+1}-1 = ...
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3answers
56 views

Converting $(1+…+n)^2*(n+1)^3$ to $(2+…+2n)^2$

I'm currently going through Calculus by Spivak by myself, and came across a proof by induction requiring to prove $1^3+...+n^3 = (1+...+n)^2$ Naturally, to prove this, I need to somehow convert ...
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2answers
39 views

Is $\sum_{i=1}^{n}\frac{f(c\frac{i}{n})}{f(c\frac{i}{n})+f(c-c\frac{i}{n})}=\frac{1}{2}(n+1)$?

So I was trying some code on Octave. The algorithm is the following $$\sum_{i=1}^{n}\frac{f(c\frac{i}{n})}{f(c\frac{i}{n})+f(c-c\frac{i}{n})}$$ for some $n\in\mathbb{N}$ and $c\in\mathbb{R}$. I ...
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1answer
40 views

How do you find the sum: $\sum_{r=1}^6 \tan^2\left(\frac{r \pi}{n}\right)$

How do you find the sum: $$\sum_{r=1}^6 \tan^2\left(\frac{r \pi}{n}\right)$$ I managed to solve this question using complex numbers so I thought I'd share the solution. If you know of any better ...
7
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2answers
91 views

How do you find the value of $\sum_{r=0}^{44} \tan^2(2r+1)$?

Problem: Find the value of $$\sum_{r=0}^{44} \tan^2(2r+1)$$ Note: The angles here are in degrees. I don't know how to solve this question because trigonometric simplifications didn't get me ...
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1answer
20 views

Problem solving summation

I know this is not a big theoratical question but I need help solving this: $$\sum_{i=1}^{n}{\sum_{j=1}^{m}{(i^2+j^3)}}$$ I need to resolve this by getting (if possible) an equation without any ...
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2answers
32 views

Are these 3 summation expressions equivalent?

$$\sum_{i=1}^{n}f(i)+k = \sum_{i=1}^{n}\{f(i)+k\} = k+\sum_{i=1}^{n}f(i) $$ I'm more confused about the working of expression: $$\sum_{i=1}^{n}\{f(i)+k\}$$ Are all the 3 expressions equivalent?
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4answers
42 views

Find upper limit of summation inequality

How is it possible, with correct calculus, to find the upper limit of a summation in an equation, this could for instance be: $\sum_{n=1}^x\frac{1}{2^n}\tag{displayed}>0.99$ How would i go about ...
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1answer
23 views

Evaluate $\sum_{p=1}^{32}(3p+2)\Bigg(\sum_{q=1}^{10}\bigg(\sin\frac{2q\pi}{11}-i\cos\frac{2q\pi}{11}\bigg)\Bigg)^{p} $

Evaluate $$\sum_{p=1}^{32}(3p+2)\Bigg(\sum_{q=1}^{10}\bigg(\sin\frac{2q\pi}{11}-i\cos\frac{2q\pi}{11}\bigg)\Bigg)^{p} $$
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1answer
26 views

Notation for writing multinomial coefficient as sum of smaller multinomial coefficients

This question is an attempt to extend the Pascal triangle's hockey stick identity to multinomial coefficients as asked in question Hockey-Stick Theorem for Multinomial Coefficients. Consider the ...
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1answer
28 views

How out of fourier line calculate below sums? [duplicate]

I put it down there (fourier line) since www page does not work. http://imgur.com/gallery/53n9qIp/new ..... 3 sums to calculate:
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1answer
122 views

Is there a way to determine $\sum_{n=0}^{\infty}\sin(2^{n}\theta )$?

Is there way to determine/simplify this infinite sum below? $$\sum_{n=0}^{\infty}\sin(2^{n}\theta )$$
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2answers
99 views

How to prove $\sum_{i=1}^ki^k(-1)^{k-i}\binom {k+1}{i} =(k+1)^k$

How to prove $\sum_{i=1}^ki^k(-1)^{k-i}\binom {k+1}{i} =(k+1)^k$ where k is a positive integer. Any hints can help.
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3answers
95 views

Prove the identity $\binom{2n+1}{0} + \binom{2n+1}{1} + \cdots + \binom{2n+1}{n} = 4^n$

I've worked out a proof, but I was wondering about alternate, possibly more elegant ways to prove the statement. This is my (hopefully correct) proof: Starting from the identity $2^m = \sum_{k=0}^m ...
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3answers
50 views

Cotangent summation (proof)

How to sum up this thing, i tried it with complex number getting nowhere so please help me with this,$$\sum_{k=0}^{n-1}\cot\left(x+\frac{k\pi}{n}\right)=n\cot(nx)$$
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0answers
67 views

find the closed form of $\sum_{k=1}^\infty\left(\frac{\sec{kz}}{k^2}\right)^2$

How to evaluate $\displaystyle\sum_{k=1}^\infty\left(\frac{\sec{(k\pi\sqrt{5})}}{k^2}\right)^2$? In general, how to find the closed form of infinite series ...
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1answer
75 views

Concrete Mathematics 2.10 - 2.13 Cannot find part of the solution

I've seen a few other questions on this section, but none match my issue, I feel like I'm missing something super obvious. Everything in here seems to make sense, I can follow along everything, except ...
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1answer
40 views

Can I take an exponent out of a sum?

For example, assuming we had a sum: $$\sum_{n=1}^m n^b \quad m,b\in\mathbb{N}$$ Is there any way to take the $b$ out of the sum? I tried taking the $\log_n$ of every value, add them together then ...
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2answers
48 views

How come it be $\frac{3}{2}A$ and not only $A$?

OK I admit I was too lazy to type this question so I took a screenshot , I got it from the site @brilliant.org where it asked in terms of $A$ what would be the 2nd summation equation ? The explained ...
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1answer
41 views

Summation: different answer than in book.

So here I have two summations, $$O_{n}=\sum_{k=0}^{n-1}f(x_{k})\cdot \Delta x \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,B_{n}=\sum_{k=1}^{n}f(x_{k})\cdot \Delta x \\ $$ ...
0
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1answer
49 views

divisibility of $\sum _{r=1}^{n} r^5$ by $\sum_{r=1}^{n}r^3$

if $$S_5=\sum_{r=1}^{n} r^5$$ and $$S_3=\sum_{r=1}^{n}r^3$$For what values of $n$ the sum $S_5$ is divisible by $S_3$. One way of approach is finding the sum $S_5$ using method of differences and ...
2
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2answers
50 views

Integral of Secant to an Even Power

As I was working on some trig integration practice problems I came across a very interesting pattern with regards to the integral of secant when its power was an even integer greater than or equal to ...
1
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1answer
21 views

Solve series with two constants and opposite exponents

How can I find the general formula for the sum of this series? $$ \sum_{i=0}^n a^ib^{n-i} $$ Where $a$ and $b$ are unrelated constants? I don't think you can split it into $ \sum_{i=0}^n a^i $ and ...
1
vote
1answer
28 views

formula to calculate average quiz score

I'm creating a Quiz web application for an assignment. There is a requirement to show a user the average score for a particular quiz (from pool of all users who have tried it before) upon successfully ...
3
votes
2answers
125 views

What is $\sum_{k=0}^\infty {k^k \over (k!)^2}$?

I know that this series converges, and the limit is approximately 3.548. But what is the exact sum, and how do you determine it?
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2answers
43 views

How to simplify this summation(s)

How can I simplfiy this summation? $$ \sum_{i = 0}^{n}(a_i * \sum_{j = i}^{n} a_j) + \sum_{k = 0}^{n} a_k $$ with $a \in \mathbb{R}$
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0answers
25 views

Quickest way to get the sum of the length of a list of vectors

sadly I'm "just" a programmer, so maybe this website is the best point for asking for advanced math "magic" :) I'm given a list of 3d-points and my task would be to get the complete length of the ...