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

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5answers
222 views

I need help with a Finite Series

Problem: Find the sum to $n$ terms of \begin{eqnarray*} \frac{1}{1\cdot 2\cdot 3} + \frac{3}{2\cdot 3\cdot 4} + \frac{5}{3\cdot 4\cdot 5} + \frac{7}{4\cdot 5\cdot 6}+\cdots \\ \end{eqnarray*} ...
5
votes
2answers
123 views

Limit involving binomial coefficient

I was trying to find the below limit. The sum can be written in a hypergeometric function but it doesn't seem to help me to find the limit. Any help will be appreciated. $$ \lim_{n \rightarrow ...
1
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4answers
119 views

Binomial Sum: Values

I need this as lemma. Regard the sums: $$S_k:=\sum_{n=0}^N\binom{N}{n}(-1)^{N-n}n^k\quad(k\in\mathbb{N}_0)$$ Then it holds: $$S_k\stackrel{k<N}{=}0\quad S_k\stackrel{k=N}{=}N!$$ How can I check ...
2
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1answer
41 views

Minkowski's inequality

Minkowski's inequality for sums states $$\left(\sum_{j=0}^\infty |a_j+b_j|^2 \right)^{1/2} \le \left(\sum_{j=0}^\infty |a_j|^2 \right)^{1/2}+\left(\sum_{j=0}^\infty |b_j|^2 \right)^{1/2} $$ for ...
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4answers
123 views
+100

Proving a function is continuous and periodic

Suppose we are given a function $$g\left ( x \right )= \sum_{n=1}^{\infty}\frac{\sin \left ( nx \right )}{10^{n}\sin \left ( x \right )},x\neq k\pi , k\in\mathbb{Z}$$ and $$g\left ( k\pi \right ...
3
votes
2answers
59 views

The sum of squares of the first $n$ natural numbers.

My basic question is this: how to find the sum of squares of the first $n$ natural numbers? My thoughts have led me to an interesting theorem: Faulhaber's formula. It is known that ...
4
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3answers
65 views

Help with a Series (Edited)

The original problem was: $$\sum_{k=0}^\infty\dfrac{k}{6k^3+13k^2+9k+2}$$ Using Partial Fractions, I resolved this into ...
3
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2answers
43 views

When are both $\sum_{n=0}^\infty \log(a_n)$ and $\sum_{n=0}^\infty a_n$ convergent?

I'm new to this site. Can someone give me some examples of when both: $$\sum_{n=0}^\infty \log(a_n)\qquad \text{ and }\qquad \sum_{n=0}^\infty a_n$$ are convergent?
7
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3answers
313 views

Solve infinite series equation with logarithmic terms.

Solve logarithmic equation: $$\frac{\log x^2}{\log^{2}x}+\frac{\log x^3}{\log^{3}x}+\cdots+\frac{\log x^k}{\log^{k}x}+\cdots=8$$ here $\log$ is assumed to have base $10$. So far I managed to rewrite ...
0
votes
1answer
25 views

Limit of summation as n goes to infinity

I am trying to solve the following: Let $q>1$ and $n \in N$. Evaluate $\lim_{n \rightarrow +\infty} \sum_{k=1} ^n \frac{k^{q-1}}{n^q + k^q}$. I understand that I need to first get the summation ...
9
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2answers
247 views
+50

How do I prove this combinatorial identity using inclusion and exclusion principle?

$$\binom{n}{m}-\binom{n}{m+1}+\binom{n}{m+2}-\cdots+(-1)^{n-m}\binom{n}{n}=\binom{n-1}{m-1}$$ Note that we can show this with out using inclusion and exclusion principle by using Pascal's Identity ...
2
votes
1answer
61 views

Coefficients of a generating function

I need a bit of help. I was solving the form of the coefficients of the generating function $\sum_{n}n^m r^n$. Then I started building the indefinite sum $\sum n^m r^n \delta n$ trough recursive ...
5
votes
3answers
75 views

Infinite sum of alternating telescoping series

I am struggling to find the sum of the following series: $$\sum_{n=1} ^{\infty} \frac{(-1)^n}{(n+1)(n+3)(n+5)}.$$ It seems as though it should be a straightforward telescoping series. I attempted to ...
2
votes
3answers
473 views

Change from product to sum

We know that : $$a \times b = \underbrace{a + a + a + ... + a}_{\text{b times}}$$ That's how we convert from a product to a sum. So what happens if we go a little further? That is : ...
3
votes
1answer
45 views

closed form for some binomial sum

I am trying to derive a closed form for the generating function of $a_n(x)=\sum_{k=0}^n \binom{n+k}{n}x^k, x>0, n\in\mathbb{N},$ i.e. for $G(z)=\sum_{n=0}^\infty a_n(x)z^n$. The only method I ...
1
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0answers
11 views

Simple question about derivative and summation

I am reading a book and I have a simple question. There is this summation: $$ A = β\sum_\textbf{x} ||\textbf{x}||^2 r(\textbf{x})$$ after this, taking the partial derivative: $$ \frac{\partial A ...
0
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2answers
37 views

Summation of powers

I have come across the following in my textbook: $$\sum_{i=0}^{20} 5^i = \frac{5^{21}-1} {4} $$ There is no explanation of how this result was achieved. Could anyone help walk-through how this would ...
2
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1answer
67 views

Summation by Parts to Evaluate $\sum_{k=1}^{\infty}(2k+1)x^{2k}$

I need to evaluate $\sum_{k=1}^{\infty}(2k+1)x^{2k}$ using the Summation by Parts (SBP) method. It is given that $0 < |x| < 1$. The notation our class uses for SBP is as follows: $$ \sum_{i} ...
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1answer
51 views

How simplify this particular sum?

Can we simplify the following sum? $$\sum_{i=1}^n \binom{n}{i} {(-1)^{i+1}\over 1-2^{-i}}$$ Thank you.
1
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1answer
101 views

Reversing the Order of Integration and Summation

I am trying to understand when we can interchange the order of Integration and Summation. I am increasingly encountering Integrals; some of which are being solved by interchanging the order of ...
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0answers
8 views

Force directed graph on a wrapped plane

I'm writing a force-directed graph where the plane is wrapped. Physics-wise this should cause the resulting forces to be an infinite sum based on each original node's distances to every recurrence of ...
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votes
3answers
63 views

Solving sum of $(-1)^n (1/2)^n$ [closed]

How to solve the following sum? $$\sum_{n=0}^k (-1)^n (1/2)^n$$
2
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1answer
35 views

Expanding summation $\sum_{i=1}^{k+1}i(i!)$

Expand the summation: $\sum_{i=1}^{k+1}i(i!)$ My solution is: $\sum_{i=1}^{k}i(i!)+k(k+1)$ But I think it is wrong. Please help. Thanks
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0answers
18 views

Is a norm of a vector a constant to a summation?

I have a simple question: if we have $$\sum_{\textbf{x}}\|\textbf{x}\|^2\, p(\textbf{x}),$$ can I treat the norm as a constant and remove it from the summation?
0
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1answer
23 views

Taking it a step further with a sum

So I was watching an "old" video from numberphile about the three square problem. https://youtu.be/m5evLoL0xwg Here is also an image: ...
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0answers
92 views

A closed form for $\int x^nf(x)\mathrm{d}x$

When trying to find a closed form for the expression $$\int x^nf(x)\mathrm{d}x$$ in terms of integrals of $f(x)$ I found that $$\int xf(x)\mathrm{d}x=x\int f(x)\mathrm{d}x-\iint ...
1
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1answer
55 views

Problem with this challenging summation

I'm having some trouble finding the summation of this series. I tried all I could, but in the end the denominator is creating problem. $$ \sum_{r=0}^{n} (-1)^r ...
1
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0answers
55 views

Solving summation; ( double sum).

I found the expression to the sum of powers long ago and ofcours I think it is true but i don't know for sure, the problem is, it's little though for me to test and try it out. Also i'd like to know ...
4
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2answers
122 views

Find the sum of first 99 terms of the sequence defined by $T_{n}=\frac{1}{5^{2n-100}+1}$

Find the sum of first $99$ terms of a sequence, where $$T_{n}=\frac{1}{5^{2n-100}+1}.$$ I need some hints on how to approach; I am unable to simplify it. Thanks.
1
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1answer
25 views

Eliminating a summation

I need to approach the new position $(x_t,y_t)$ at moment $t$ of a moving object at $(x_0,y_0)$ given its horizontal velocity $vx_0$, its vertical velocity $vy_0$ and some constant resistance $r$ that ...
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0answers
29 views

Checking if the sum of a set of functions can ever be zero

Given a set of elementary functions $f_1(x), f_2(x), f_3(x), ..., f_n(x)$ that are known to either be zero at some point, or always nonzero, it is trivial to check if the product of all these ...
0
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1answer
47 views

Combinatorial proof for $\sum_{k = 0}^n \binom {r+k} k = \binom {r + n + 1} n$ [duplicate]

I'm trying to figure out a combinatorial proof for: $$\displaystyle \sum_{k \mathop = 0}^n \binom {r+k} k = \binom {r + n + 1} n$$ I've tried the committee counting thing, but that didn't work.
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0answers
20 views

Fourier cosine transformation

Good day! I'm studying right now some transformation and I encountered the following equation: $$(2\pi)^{-n/2} \int_{-\infty}^\infty\cdots\int_{-\infty}^\infty \exp\left(-\frac{1}{2} ...
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2answers
29 views

Is there a shortcut to summing fractional powers?

I'm trying to solve a problem where, incrementally, each step sums a particular value. The value is plagued with an ugly fractional power. Is there a shortcut to something like $$\sum_{i=0}^n ...
0
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1answer
20 views

Special Form of Combination - Formula Verification

Giving abc how many combination that involves a ? answer is a ab ac abc equal to 4 I came up with the following formula but I would like to know of its correctness plus if there is simpler form ...
1
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1answer
26 views

Question about the Cauchy Product and how it is done

Lets say we have the following: $$ \sum_{k=0}^\infty z^k \sum_{j=0}^k \frac{1}{j!(k-j)!} B_{k-j}^f(x) \frac{d^{j}}{dx^{j}}[a_k(x)] $$ Would it be correct to say that: $$ \sum_{k=0}^\infty z^k ...
2
votes
3answers
43 views

Prove that for any positive integer $n$ and $d$, $\sum_{k=0}^d 2^k\log_2(\frac{n}{2^k})=2^{d+1}\log_2(\frac{n}{2^{d-1}})-2-\log_2{n}$

I could prove it by induction, but I need to see how I might have discovered it for myself (cause that's what's gonna be on exam).
4
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1answer
101 views

Formula for $\sum\limits_{j=1}^{m-1}\frac{1}{\sin^{2p}(\frac{j\pi}{m})}$

Let $m\geq 2$ be an integer, then there is the well known formula $$\sum\limits_{j=1}^{m-1}\frac{1}{\sin^2(\frac{j\pi}{m})}=\frac{m^2-1}{3},$$ I'm interested in similar equations for the following ...
1
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3answers
81 views

Induction Proof: $\sum_{k=1}^n k^2$

Prove by induction, the following: $$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}6$$ So this is what I have so far: We will prove the base case for $n=1$: $$\sum_{k=1}^1 1^2 = \frac{1(1+1)(2(1)+1)}6$$ We ...
4
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1answer
68 views

Combinatorial interpretation of identity

I recently came across the identity $$\sum_{k=0}^m\dbinom{m}{k}\cdot \frac{(-1)^k}{n+k+1}=\dfrac{n!\cdot m!}{(n+m+1)!},$$ while working on evaluating $$\int_0^1 x^n(1-x)^m\, dx.$$ I ended up ...
0
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1answer
128 views

What is the formula for summation of $n^n$? [closed]

How should I calculate: $$1^1+2^2+3^3+\dotsb +n^n$$ What is the formula for this submission?
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4answers
72 views

Changing limits of a summation?

I've been doing a question which asks you to prove the following is true: $$ \sum_{r=1}^{n}r(r+1)(r+2)(r+3)(r+4)$$ $$=\frac{1}{6}n(n+1)(n+2)(n+3)(n+4)(n+5) $$ which is realtively straightforward using ...
3
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1answer
29 views

Summation functions for wall clock, 10AM, 11AM and 12PM tips needed

For a recreational purposes I'm fine tuning my wall clock sheet and like to ask about tips how to esthetically modify the summation function for 10, 11 and 12. Below is the image of the final result: ...
0
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1answer
47 views

Sequence Series/Summation Problem

I was given the below formula to solve: $$ \sum_{k=1}^{40}k^{2k} $$ I'm not sure how to approach the problem considering $k$ isn't a constant to obtain any values. Any hints would be appreciated.
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2answers
34 views

Combinatorial proof for summation of powers of two

I apologise if this has been posted before, but I've been poring over this problem for days now and just can't seem to get it. I'm looking for a combinatorial proof for: $2^n - 1 = 2^0 + 2^1 + 2^2 + ...
1
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1answer
51 views

Verify the given error 0f 0.1813 is accurate by approximating $\sum_{n=1}^{\infty} \frac{1}{n^2}$ by the first 5 terms

Verify that the error in approximating the first 5 terms of the series $\sum_{n=0}^{\infty} \frac{1}{n^2}$ is approximately 0.1813. The formula I have to find the error is if $S=\sum_{n=0}^{\infty} ...
0
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3answers
54 views

Sums of infinite series

Determien the sums of the following series'. 1:$\sum_{k=0}^\infty \frac{2+(-1)^k}{3^k}$ 2:$\sum_{k=0}^\infty (\frac{1}{n}-\frac{1}{n+2})$ 3:$\sum_{k=0}^\infty \frac{1}{4k^2-1}$ ...
2
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2answers
136 views

Euler-Maclaurin Summation

Using EM summation formula estimate $$ \sum_{k=1}^n \sqrt k $$ up to the term involving $\frac{1}{\sqrt n}$ My attempt is $$ \sum_{k=1}^n \sqrt k = \frac{2 \sqrt{n^3}}{3} -\frac{2}{3} + \frac 1 ...
0
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6answers
144 views

Find the sum $1^2+3^2+…+(2n-1)^2$ without induction

I tried with $$(2n-1)^3-(2n-2)^3=12n^2-18n+7$$ Now, forming partial sums for $n=1,2,...$ $$(1^3-0^3)+(3^3-2^3)+...+((2n-1)^3-(2n-2)^3)=12(1^2+...+n^2)-18(1+...+n)+7n$$ How to express ...
0
votes
3answers
61 views

Simplifying $\sum_{i=1}^n {1\over i(i+1)}$ to ${n\over n+1}$

I need to get from this: $$\sum_{i=1}^n {1\over i(i+1)}$$ to: $${n\over n+1}$$ or: $${1 - {1\over n+1}}$$ I have tried looking for sums identities with fractions, using WolframAlpha.com (that's how I ...