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

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Sum of the series where exponents are in geometric progression

Consider the following series: $x^2, x^4, x^8, \dots, x^{2^n}$ Since the series grows very very quickly, the total sum is dominated by the last term and is less than $2 \cdot x^{2^n}$. Is there a ...
2
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1answer
63 views

Sum of arithmetico-geometric series

Could use help trying to find the following sum of series $$ \sum_{n=1}^N r^n\sqrt{a + nd} $$ I have no clue where to begin on this one. Ideally would like solution for all $ r $ but if it helps ...
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3answers
37 views

Closed form expression of a summation

My prof started out with the following summation: \begin{equation} \sum_{i=0}^{k}i = \frac{k(k+1)}{2} \end{equation} Which is all fine and dandy, however the summation we want to find the closed form ...
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1answer
32 views

Proof that the squareroot of the mean of the squares is allways greater or equal than the mean of weighted values

I couldn't find a better title, but basically you have given some values $x_1...x_n$ and some weights $p_1...p_n$ ($x_n\in\mathbb{R}$ and $p_n\in[0,1]$, also $p_1+...+p_n=1$). You now calculate the ...
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3answers
89 views

The sum of the first $n$ squares $(1 + 4 + 9 + \cdots + n^2)$ is $\frac{n(n+1)(2n+1)}{6}$ [duplicate]

Prove that the sum of the first $n$ squares $(1 + 4 + 9 + \cdots + n^2)$ is $\frac{n(n+1)(2n+1)}{6}$. Can someone pls help and provide a solution for this and if possible explain the question
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1answer
32 views

Is it possible to minimize this summation?

I am given the following information: $$\sum_{i=1}^{n}a_i = 1$$ $$\forall i \in n \quad a_i > 0$$ I would like to minimize the following summation: $$\sum_{i=1}^{n}a_i^2$$ I don't really know where ...
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1answer
94 views

Confusion about $\sum _{n=1}^{\infty}{(1+2+…+n)}$ [closed]

$\sum _{n=1}^{\infty}{(1+2+ ... +n)}$ I came across this example and it seems to be trivial. However, I am not particularly sure what it is going to equal if $n$ goes not to $\infty$, but to $n=2$, ...
2
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2answers
88 views

Calculating $\sum_{n=1}^\infty {\frac{1}{2^nn(3n-1)}}$

I'd appreciate any help, I know it has something to do with the geometric series but I still can't figure out how. I thought about integration but couldn't find a way to do it. $$\sum_{n=1}^\infty ...
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1answer
24 views

Test the convergence.

Test the convergence of $\left (a_{n} \right)$ if $$a_{n}=\sum_{k=1}^{n}\left( \sqrt[\left \lfloor \frac{m}{4} \right \rfloor]{k^2+1} -\sqrt[\left \lfloor \frac{m}{4} \right \rfloor]{k^2-1} \right), ...
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0answers
32 views

Prove:$\sum\limits_{k=0}^{n}\frac{(-1)^k{n\choose k}}{k+1}=\frac{1}{n+1}$ [duplicate]

Prove:$\sum\limits_{k=0}^{n}\frac{(-1)^k{n\choose k}}{k+1}=\frac{1}{n+1}$ I tried to prove the equation by induction, but can't find the relation between $\sum\limits_{k=0}^{m}\frac{(-1)^k{m\choose ...
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1answer
39 views

Squared dobule sum expression?

Is there a way to get expression for squared double sum? $$\left(\sum\limits_{i=1}^n \sum\limits_{j=1}^n a_i a_j\right)^2 = \left(\sum\limits_{i=1}^n \sum\limits_{j=1}^n a_i ...
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1answer
30 views

How to pove this identity with sum

show that $$\sum_{k=1}^{n}\left(\binom{n}{k}\dfrac{\Gamma{(k+1)}}{\Gamma{(k+1-a)}}x^{k-a}\right)=\dfrac{\Gamma{(n+1)}}{\Gamma{(n+1-a)}}(1+x)^{n-a}$$
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0answers
35 views

How do I find the sum of this series? [duplicate]

This is part of a homework assignment so I'm only looking for a hint on how to solve this. The question is to find the sum of the series $\sum\limits_{n=1}^\infty \frac{(-1)^{n-1}}{(2n-1)3^{n-1}}$ I ...
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5answers
275 views

Power series summation [closed]

Trying to find the sum of the following infinite series: $$ \displaystyle\sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}}{(2n-1)3^{n-1}}$$ Any ideas on how to find this sum?
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2answers
69 views

Computing $\sum{\frac{1}{m^2+n^2}}$

I want to prove that $\sum_{1\leq m^2+n^2\leq R^2}{\frac{1}{m^2+n^2}}=2\pi\log R+O(1)$ as $R\rightarrow\infty$. For this, I'm trying to approximate the sum by using the integral $\int_{1\leq r\leq ...
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1answer
31 views

Can the sum $\sum_{k=1}^\infty (1/k)^{3/2}\sin(kx)$ be evaluated using Fourier series or otherwise?

I have to compute this sum, and I was wondering if it can be evaluated using Fourier series. It seems familiar to me but have forgotten the Fourier tricks I used in the past, so time for revision. ...
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0answers
32 views

How to find asymptotic sum of highly oscillatory series?

I have a sum given by, $$(1) \quad | S_w=\sum_{n=1}^{w} 3^{n/2} \cdot \sin(3^n \cdot t) |$$ How do I find the value of $(1)$ asymptotically? I can guess, using knowledge about the fractal dimension ...
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3answers
282 views

What is the sum of the series $x^{n-1} + 2x^{n-2} + 3x^{n-3} + 4x^{n-4} + \cdots + nx^{n-n}$?

$x^{n-1} + 2x^{n-2} + 3x^{n-3} + 4x^{n-4} + \cdots + nx^{n-n}$ I'm not sure if this can even be summed. Any help is appreciated.
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2answers
46 views

Discrete Calculus — Summing $\sum _{k=1}^{n+1} \frac{k}{(n+1)^k (-k+n+1)!}$

$$\sum _{k=1}^{n+1} \frac{k}{(n+1)^k (-k+n+1)!}$$ Mathematica gives that the answer is quite surprisingly the reciprocal of $n!$. The first way I thought of trying to prove this was by induction, ...
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0answers
31 views

Cumulative distance formula

I was asked from Physics SE to move this question here. I am working with position recordings over time. Basically, I record the position of animals in a treadmill. They can go forwards or backwards. ...
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1answer
35 views

The sum of binomial coefficients $\sum_{i=1}^n \binom{i}{2}= {n+1 \choose 3}$

Prove by induction: $$\sum_{i=1}^n \binom{i}{2}= {n+1 \choose 3}$$ I already know that: $$\sum_{i=1}^n \binom{i}{2} = {i+1 \choose 2+1}$$ And the LHS is now equal: $$\sum_{i=1}^n \binom{i}{2} + ...
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2answers
56 views

log of summation expression

I am curious about simplifying the following expression: $\log \left(\sum_\limits{i=0}^{n}x_i \right)$ Is there any rule to simplify a summation inside the log?
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1answer
139 views

Finding $\binom{999}{0}-\binom{999}{2}+\binom{999}{4}-\binom{999}{6}+\cdots +\binom{999}{996}-\binom{999}{998}$

How to find this alternating sum of binomial coefficients? $$\binom{999}{0}-\binom{999}{2}+\binom{999}{4}-\binom{999}{6}+\cdots +\binom{999}{996}-\binom{999}{998}$$
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2answers
52 views

How to calculate this infinite sum?

$$ \sum_ {n=0}^\infty \frac {1}{(4n+1)^2} $$ I am not sure how to calculate the value of this summation. My working so far is as follows: Let $S=\sum_ {n=0}^\infty \frac {1}{(4n+1)^2}$. ...
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2answers
49 views

How to calculate recursive series?

$r_1 = r_0 (1 + e)$ $r_2 = r_1(1 + 2e)$ $r_3 = r_2(1 + 3e)$ $r_n = r_{n-1}(1 + ne)$ Can I get $r_n$ in closed form (a single formula) in terms of $r_0$ and $e$ ? Also, as n tens to infinity what ...
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1answer
17 views

Eigenvalues versus moments. Identity regarding multinomials.

Let $\left\{ c_k\right\}_{k=1}^N$ be some numbers. We can think of them as being eigenvalues of a $N$-dimensional matrix. Now we take a positive integer $p$ and we consider a following quantity: ...
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30 views

derivative of log gamma under summation

I have a problem with taking derivative of this equation with respect to $\alpha_i-1$ $$\frac{\partial}{\partial(\alpha_i-1)} \left[\log \Gamma\left(\sum^k_{i=1} \alpha_i\right) − \sum^k_{i=1} \log ...
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0answers
119 views

Summation of the absolute value of the variable

The summation of cosine $\sum_{k=1}^N \cos (k x)$ is well known (for example, see the previous question here) and is called Lagrange's trigonometric identity. Is it possible to construct a similar ...
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23 views

Can the following sum involving double factorial be simplified?

I have the following sum: $$\sum_{k=2}^{n-1}\frac{(n-k)(2k-3)!!}{a^k}$$ where $p!!$ is the double factorial defined as $$p!!=\left\{\begin{array}{} p\cdot (p-2)\cdots 3\cdot 1, & \mbox{if $p$ is ...
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1answer
15 views

Proving absolute convergence by proving limit via power condition

I was given the following premise: If $\lim\limits_{x \to 1} \frac{|a_{n+1}|}{|a_n|} = r<1$, then $\sum a_n$ is absolutely convergent by the following: Show $\sum (r+\epsilon)^n < \infty$ I ...
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2answers
32 views

Prove $ \sum_{k=0}^n k4^k = \frac 49((3n-1)4^n + 1) $ by induction

Prove that for every position integer $n$ that $$ \sum_{k=0}^n k4^k = \frac 49((3n-1)4^n + 1) $$ Proof: Let $P(n)$ denote the above statement. Base case: $n=1$ : Note that $$ \sum_{k=1}^1 k4^k = ...
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1answer
30 views

Proving multiplying a 2nd degree term by a conditionally convergent series converges absolutely?

I was asked to prove that if $\sum a_n$ converges conditionally $\Rightarrow$ $\sum n^2 a_n$ diverges. The proof prompt hints at using proof by contradiction. So far, I have shown that $a_n$ ...
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1answer
5 views

Minus a Value from a Sum

Think I'm being really stupid. I have the following sum: $\sum\limits_{n=1}^r x^n$ which is close to a geometric series with the limits being $0$ and $r-1$ (ideally I want to be using that form) I ...
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3answers
50 views

Finite Series Identity

EDIT: I stated a FALSE question, because I understood bad. Please read again. How to prove that $$\sum_{n=1}^{\infty} \frac{1}{n(n+k)} = \frac{1}{k} \sum_{n = 1}^{k} \frac{1}{n}$$ using only basic ...
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2answers
40 views

Infinite sum to fraction

I have the following infinite sum: $$\sum_{n=0}^{\infty}(n+1)^2 \cdot z^n$$ Could you help me how can I convert it to the fraction form? $$-\frac{z(z+1)}{(z-1)^3}$$ (when $|z| < 1$)
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1answer
25 views

Combinatoric Summation

How do i evaluete : $\sum_{j=0}^{n}\left \{ \binom{n}{j}\; \sum_{i=0}^{j}\binom{j}{i}8^{i} \right \}$. I know $\sum_{k=0}^{n}\binom{n}{k}=2^{n}$ But how about the rest?
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14 views

Polynomials of power $N$

Let $f(x)= \sum\limits_{i=1}^{r}a_ix^{b_im+c_in}$ be a polynomial, where $b_i,c_i\in\mathbb N$, $a_i\in \mathbb Q$ and $n,m \in \mathbb Z$. Is there any integer $N$ such that $(f(x))^N= ...
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30 views

Solve the following sequence equation

$${1\over4⋅8} +{1\over12⋅8}+ {1\over6⋅12}+...+{1\over4n(4n+4)}+{1\over16(n+1)}={1\over16} $$ I have tried finding a reccurence formula or,at least ,prove the given one(the last two terms),but i ...
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1answer
75 views

Sum up a function series $f(1/9)+f(2/9)+\dots+f(26/9)$ for $f(x)=\frac{9^{x}}{9^{x}+27}$

Given $f(x)=\dfrac{9^{x}}{9^{x}+27}$. Find: $$S=f\left(\frac{1}{9}\right)+f\left(\frac{2}{9}\right)+\dotsb+ f\left(\frac{26}{9}\right)$$ Teacher did not allow us to use calculator...Use sigma ...
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1answer
71 views

Evaluating a double sigma

Evaluate $$\sum_{m=0}^{\infty} \sum_{n=0}^{\infty}\frac{m!n!}{(m+n+2)!}$$ How do I start with the problem? Infinite sum of factorials?
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1answer
29 views

Estimate for sum of negative powers of primes [duplicate]

Specifically, for $a \in (0,1)$, I am interested in the sum $$\sum_{p\leq n} \frac{1}{p^a} $$ as $n$ grows.
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1answer
77 views

How is $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}= \log 2?$ [duplicate]

How is $$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}= \log 2?$$ I haven't done sequences in a long time, therefore proving this seems almost impossible. How is this sum gotten. Help very much ...
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0answers
40 views

How to prove this determinant equation?

Given two matrices $A \in \text{Mat}_{n\times m}$ and $B \in \text{Mat}_{m\times n}$ for $m \geq n$, I need to prove this: $$\det(AB) = \sum\limits_I \det(A_I)\det(B_I),$$ where I passes (?) all ...
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2answers
51 views

Simplification of $\sum_{k=1}^n \frac{1}{4k^2-1}$

So I want to simplify this expression: $$\sum_{k=1}^n \frac{1}{4k^2-1}$$ and Wolfram Alpha tells me it can be simplified to two forms: $$\frac{n}{2n+1}$$ and $$\frac{1}{2}-\frac{1}{2(2n+1)}$$ The ...
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1answer
58 views

How to find the sum: $ \sum_{i = 0}^n i^{1/5} $

Given the sum: $$ \sum_{i = 0}^n i^{1/5} $$ How to find $A$ in: $$ \sum_{i = 0}^n i^{1/5} = A + O(\frac1{n^6}) $$ I tried to use Euler–Maclaurin formula and obtained numbers that confused me?
2
votes
3answers
32 views

Summation of $\frac{\cos n \theta}{2^n}$

I would like to compute the following sum: $$\sum_{n=0}^{\infty} \frac{\cos n\theta}{2^n}$$ I know that it involves using complex numbers, although I'm not sure how exactly I'm supposed to do so. I ...
1
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1answer
34 views

Summation with a variable as the upper limit

$$\sum_{n=1}^m \frac{n \cdot n! \cdot \binom{m}{n}}{m^n} = ?$$ My attempts on the problem: I tried writing out the summation. $$1+\frac{2(m+1)}{m} + \frac{3(m-1)(m-2)}{m^2} + \cdots + ...
0
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0answers
25 views

Show that there exist $a_1, a_2, \dots, a_p$ such that $a_i + a_j$ are pairwise distinct

Let $p$ be an odd prime. Show that there exist $p$ positive integers $a_1, a_2, \dots, a_p$ such that $a_i \leq 2p^2, \forall i$ and the sums $a_i + a_j$ $(1\leq i < j \leq p)$ are pairwise ...
0
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1answer
24 views

Sum of $\sum_{n=1}^{N} \cos(kx-\omega t +n\theta)$

I am familiar with Lagrange's trigonometric identities which are $$\begin{align} \sum_{n=1}^N \sin (n\theta) & = ...
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0answers
15 views

Sum of finite number of terms of the series $\sum\limits_{L=2}^{L_{max}}\frac{1}{e^{i \phi / L} -1}$

Good day everyone. Are there any chances to get a compact formula for the following sum of finite number of terms? $$\sum\limits_{L=2}^{N} \dfrac{1}{e^{ \frac{i \varphi}{L}} -1}$$ N and $\varphi$ ...