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

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7
votes
2answers
141 views

How to count matrices with rows and columns with an odd number of ones?

I proved that $\displaystyle \left(\sum_{k\, \rm odd}\binom{m}{k}\right)^{n-1}=\left(\sum_{k\;{\rm odd}}\binom{n}{k}\right)^{m-1}$ by counting matrices of size $n\times m$ with entries in $\{0,1\}$ ...
3
votes
2answers
78 views

Euler-Maclaurin summation for $e^{-x^2}$

I want to approximate the sum $$\sum_{k=0}^\infty e^{-k^2}$$ using the Euler-Maclaurin formula $$\sum_{k=0}^\infty f(k) = \int_0^\infty f(x) \, dx + \frac{1}{2}(f(0) + f(\infty)) + ...
5
votes
1answer
89 views

Proving that $\sum_{k=0}^n\frac{1}{n\choose k}=\frac{n+1}{2^{n+1}}\sum_{k=1}^{n+1}\frac{2^k}{k}$

I want to prove for any positive integer $n$, the following equation holds: $$\sum_{k=0}^n\frac{1}{n\choose k}=\frac{n+1}{2^{n+1}}\sum_{k=1}^{n+1}\frac{2^k}{k}$$ I tried to expand $2^k$ as ...
0
votes
2answers
28 views

Compute $\sum_{i=0}^{2n} (-3)^i$ by splitting the series into two parts.

Compute $\sum_{i=0}^{2n} (-3)^i$ by splitting the series into two parts. How do I split it into two parts? All I can tell so far is that the sum is going to be a positive number (probably) because ...
0
votes
2answers
51 views

What is the sum of $1^3q + 2^3q^3 + 3^3q^3 +\cdots+ n^3q^n$?

What is the sum of $1^3*q + 2^3*q^2 + 3^3*q^3 +...+ n^3*q^n ?$
-1
votes
2answers
24 views

Compute the sum $\sum_{i=0}^n 5^{i+1}-5^i$

Compute the sum: $$\sum_{i=0}^n 5^{i+1}-5^i$$ with the hint, "start by writing out (and expanding) the sum." So I did and got $$4 + 20 + 100...$$ with the appearance of going to infinity. Is ...
4
votes
1answer
108 views

Using the Parseval Identity to compute $ \sum_{n=1}^{+ \infty} \frac{1}{(4n^2-1)^2}$

Parseval's Identity: For continuous $f: [- \pi , \pi] \to \mathbb{R}$ $$ \sum_{n=- \infty}^{+ \infty} |c_n|^2 = \frac{1}{2 \pi} \int_{ - \pi}^{ \pi} |f(x)|^2dx, \text{ where } c_n = ...
4
votes
5answers
175 views

What is the sum of $1^4 + 2^4 + 3^4+ \dots + n^4$ and the derivation for that expression

What is the sum of $1^4 + 2^4 + 3^4+ \dots + n^4$ and the derivation for that expression using sums $\sum$ and double sums $\sum$$\sum$?
1
vote
3answers
34 views

Using $S_n = \sum_{k=1}^{n}H_k$ where $H_k$ are the harmonic numbers, show $S_n = (n+1)H_n - n$ [duplicate]

The question: Using $S_n = \sum_{k=1}^{n}H_k$ where $H_k$ are the harmonic numbers, show $S_n = (n+1)H_n - n$. So far I have $S_n = \sum_{k=1}^{n} H_k = \sum_{k=1}^{n} ...
0
votes
1answer
36 views

What is the infinite sum of regular numbers?

What and is the infinite sum of regular numbers? $\sum 1/r$ where r is regular number. Upto 10 digit accurate. Thanks Edit. http://oeis.org/A003592
0
votes
1answer
14 views

Find required increase per day. Find X

So I need help with formula. We have a price that is published every day for example today's was $23995. We also have a month to date, which is obviously the average of all the daily prices this ...
1
vote
1answer
25 views

Expected value of a sum starting at a value given through a random variable

I've got a question concerning the expected value of a sum which starts at a certain value given through a random variable. More precisely: $$G(n):=P[X \geq K]$$ where $X \in Bin(n,p)$ and $K$ is ...
0
votes
2answers
29 views

Lagrange polynomials sum to one

I've been stuck on this problem for a few weeks now. Any help? Prove: $\sum_{i=1}^{n}\prod_{j=0,j\neq i}^{n}\frac{x-x_j}{x_i-x_j}=1$ The sum of lagrange polynomials should be one, otherwise affine ...
0
votes
1answer
15 views

What steps and properties are involved in this Summation simplification?

I have reference material that shows how to get from step 2 to 3 by recognizing the harmonic series. But how do they get from step 1 to 2?
0
votes
1answer
72 views

closed form for $\binom{n}{0}+\binom{n}{3}+\binom{n}{6}+…+\binom{n}{n}$ [duplicate]

closed form for $$\binom{n}{0}+\binom{n}{3}+\binom{n}{6}+...+\binom{n}{n}$$ I tried to solve it by : $$\binom{n}{0}+\binom{n}{3}+\binom{n}{6}+...+\binom{n}{n}=\sum_{k=0}^{n/3}\binom{n}{3k}$$ ...
0
votes
2answers
56 views

Sum of $k^4$ from $0$ to $n$

How can I find this summation? I started by expanding $(k+1)^5$ and setting the summation of both equal to each other. There is some cancellation but I don't know what to do afterwards.
0
votes
3answers
72 views

Easy Math question : Sum of squares

How to sum $2^2 + 4^2 + 6^2 + \dots + (2n)^2$ upto n terms. Also what if we have to sum $1^2 + 3 ^2 + \dots + (2n+1)^2$ up to n terms. I am new to this topic so please answer in a simple manner
0
votes
1answer
26 views

Close form of a power series starting at $n=2$

This is the power series I am looking at $\sum_{n=2}^{\infty}{n(n-1)z^n}$. I want to find the closed form of this power series. This is my approach, if I divide the power series by $z^2$, then I ...
0
votes
1answer
42 views

Prove there is a subsequence $(a_{nk})_{n=1}^\infty$ such that $\Sigma^{\infty}_{k=1} a_{nk}$ converges.

Hey everyone this was give as a practice problem and i'm having trouble, any help is appreciated Let $(a_n)_{n=1}^\infty$ be a sequence such that $\displaystyle \lim_{n \rightarrow \infty} {a_n} = ...
0
votes
1answer
58 views

Inequality Question about Converging Sum

This is from the UPenn prelim questions. http://hans.math.upenn.edu/amcs/AMCS/prelims/prelim_review.pdf (I asked the question before, and there was no answer, so I am asking it again.. I'm not sure ...
0
votes
1answer
23 views

Using induction to prove $\sum\limits^n_{k=1} 9^k = 0.5 \cdot \sum\limits^{2n}_{k=1} (-1)^k \cdot 3^{k+1}$

$$\sum^n_{k=1} 9^k = 0.5 \cdot \left[\sum^{2n}_{k=1} (-1)^k \cdot 3^{k+1}\right]$$ I have tested both with a python script and it seems to be correct. For the life of me, I am unable to unwind the ...
0
votes
1answer
54 views

What is mathematical term to describe this confusion?

This is in reference to a question on stackoverflow - http://stackoverflow.com/questions/22445470/getting-more-data-while-converting-data-int-to-float-and-doing-division-and-mult#22445470 The ...
1
vote
1answer
38 views

What is the reasoning behind ways of splitting up this summation sign?

Some context: I've been studying Chebyshev's $\psi$ - function, which claims that $\psi(x) = \sum_{n \le x} \Lambda(n) = \sum_{p^k \le x} \log p$ where $p$ is prime and $\Lambda(n)$ is the von ...
3
votes
2answers
124 views

Summation involving totient function: $\sum_{d\mid n} \varphi(d)=n$ [duplicate]

Prove that:$$\sum_{d\mid n} \varphi(d)=n$$ Where $\varphi(n)$ denotes the number of positive integers $m$ less than or equal to $n$ such that $\gcd(m,n)=1$ I am lost here, any help would be ...
0
votes
1answer
40 views

Why does this equation holds?

Could anyone tell me why following equation holds? $ \sum_{n \geq 0} x^n \sum_{i \geq 0} \binom{i}{n-i} = \sum_{i \geq 0} x^i \sum_{n \geq 0} \binom{i}{n-i} x^i$
4
votes
2answers
62 views

Is $f(x)=\sum_{k\in\mathbb N}\frac1k\sin\frac x{2^k}$ bounded?

$$f(x)=\sum_{k\in\mathbb N}\frac1k\sin\frac x{2^k}$$Is this function bounded? So obviously this converges because $|\frac1k\sin\frac x{2^k}|<|\frac x{2^k}|$ and $\sum\frac x{2^k}$ converges by ...
6
votes
2answers
148 views

Find the integral part of $\sum_{i=2}^{10000}\frac1{\sqrt{i}}$

$$A = \frac1{\sqrt{2}}+\frac1{\sqrt{3}}+\cdots+\frac{1}{\sqrt{10000}}$$ Find $\lfloor A\rfloor$ where $\lfloor x\rfloor$ is the greatest integer less than, or equal to $x$ I got stuck on this, so ...
0
votes
0answers
22 views

Fast way to update sum of logs after each input is shifted by the same value

Suppose you have a sum of $n$ positive, increasing elements: $$\textrm{sum}_{\ 1} = \ln(x_1)+\ln(x_2)+\dots+\ln(x_n)$$ Where $x_1,\dots,x_n \in \Bbb R^+$. For a value of $C > -x_1$, where $x_1$ ...
1
vote
1answer
67 views

Evaluating Summations

I want to calculate the infinite sum: $$\sum^{\infty}_{k=1} \frac{e^{-5}5^{2k-1}}{(2k-1)!}$$ I know this series converges by the ratio test. So I must compute the limit: $$\lim_{n \to \infty} ...
3
votes
1answer
122 views

Evaluate $\sum_{k = 0}^{n} {n\choose k} k^m$

So, I wonder what is the evaluation of $$\sum_{k = 0}^{n} {n\choose k} k^m\text{,}\qquad (*)$$ where $m,n\in \mathbb{N}$. One of my tries: knowing that $$k^m = \sum_{j = 0}^{m}\text{S}(m,j)\cdot ...
2
votes
3answers
105 views

Proof without words for $\sum_{i=0}^\infty(-1)^i\frac{1}{2i+1}$

$$\sum_{i=0}^\infty(-1)^i\frac{1}{2i+1}$$ $$1-\frac13+\frac15-\frac17+\frac19-\cdots=\frac\pi4$$ Does anyone know of a proof without words for this? I am not looking a for a just any proof, since I ...
0
votes
1answer
27 views

Understanding relation between Product and Summation Notation

So I am given the following: $n = \sum_{i=1}^{k}m_{i}$ I am also given $x = \sum_{i=1}^{k}log(m_{i}) = log\prod_{i=1}^{k}m_{i}$ I was only given the first part, however I believe that is a ...
0
votes
1answer
41 views

Question on the Koebe-Bieberbach Theorem

Assume $f$ is injective and that $f(0) = 0$ and $f'(0) = 1$. The theorem states that $\exists r >0$ such that $D_r(0) \subset f(\mathbb D)$ and, at best, $r=1/4$ ($\mathbb D$ is the unit disc). ...
2
votes
4answers
254 views

Why is my series wrong?

Why is this series wrong and how does it differ from this other one? We had to find the general term for the series: $ 1/3+2/9+1/27+2/81+1/243+2/729+\ldots $ where the index begins at $n=1$ So I came ...
5
votes
1answer
241 views

Are these two binomial sums known? Proven generalization to the Hockey Stick patterns in Pascal's Triangle

English translation. You can see the original - deprecated - in Portuguese here Hi, I arrived at a generalization for the Hockey Stick Patterns, from our beloved Pascal's Triangle. This ...
0
votes
0answers
32 views

Approximation for the logarithm of a summatory

I would like to find an approximation for: $$ \log \left(\sum_{i=1}^{N} a_i\exp(-b_i^2)\exp(-c_i^2)\right) $$ with $$ a_i = \frac{1}{\sqrt{(e^2 + e_i^2)(g^2 + g_i^2)}} \\ b_i = \frac{b-d_i}{2(e^2 + ...
2
votes
0answers
32 views

Sum of Binomials times Logarithms

Is there a closed-form expression or a very good approximation for $$ \sum_{i=0}^n \binom{n}{i} \log (i+1) \,? $$ If the summands alternate, then there is a very close approximation, yet it feels ...
2
votes
3answers
50 views

A sum that scales as the square root of the summands

Is there some (edit! analytic) expression $h(x)$ such that the sum $$\sum_{i=1}^n h(i)$$ scales as $O\left(n^\frac{1}{2}\right)$? Regarding the (40) comments under Sabyasachi's accepted answer: When ...
1
vote
1answer
29 views

O(1) Exponential summation

Is there an O(1) (uses a function instead of summation/for loop notation) way to calculate $$ \sum\limits_{i=0}^n x^i $$ ...
0
votes
3answers
75 views

Proofs Homework Help

I have been struggling with my proofs homework this week and would greatly appreciate any help. Prove, disprove, or give a counterexample: Suppose $f:X\to Y$, $A\subseteq Y$, $B\subseteq Y$ and ...
2
votes
5answers
104 views

How find this sum $f(n)=\sum_{i=1}^{n}\dfrac{\binom{n}{i}}{i}$

Find this sum closed form $$f(n)=\sum_{i=1}^{n}\dfrac{\binom{n}{i}}{i}$$ My idea: since $$\dfrac{1}{i}=\int_{0}^{1}x^{i-1}dx$$ so ...
0
votes
1answer
34 views

Summation notation rule

Sorry if this sounds elementary, but I have problems with the following in a text I am reading: $$ \left(\sum_{k=0}^{n} C_k\phi_k(x)\right)^2 = \sum_{k=0}^{n}\sum_{l=0}^{n}C_k ...
0
votes
3answers
81 views

Derive Closed form sum of N^2

Can anyone explain to me how you would derive this ? I have this question asked in a CS class and can't figure out how to derive it. it has to be derived as you would with sum of N ex ...
12
votes
3answers
178 views

Prove $1^2+2^2+\cdots+n^2 = {n+1\choose2}+2{n+1\choose3}$

Prove that: $$ 1^2+2^2+\cdots+n^2 = {n+1\choose2}+2{n+1\choose3} $$ Now, if I simplify the right hand combinatorial expression, it reduces to $\frac{n(n+1)(2n+1)}{6}$ which is well known and can be ...
3
votes
2answers
66 views

how to prove that $\sum_{k=1}^{m}k!k=(m+1)!-1$ without induction?

how to prove that $$\sum_{k=1}^{m}k!k=(m+1)!-1$$ without induction ? my only try is to put $k!=\Gamma(k+1)$ then use geometric series with some steps but I got complicated integral If any one can ...
1
vote
4answers
80 views

Evaluating $\lim_{n\to\infty} \left({1\over1\cdot2\cdot3}+{1\over2\cdot3\cdot4}+\cdots+{1\over(n-1)\cdot n\cdot(n+1)}\right)$

The original question was to find $L=\displaystyle\lim_{n\to\infty}\sum_{k=1}^na_k$ where $a_n=\displaystyle{n\over(1+2+\cdots+(n-1))(1+2+\cdots+n)}$, which I managed to get down to evaluating the ...
1
vote
2answers
45 views

Minimizing $\sum_{i=1}^n \frac{x_i^2}{w_i}$ subject to $\sum_{i=1}^n x_i=1$

Minimize $\displaystyle\sum_{i=1}^n \frac{x_i^2}{w_i}$ subject to $\displaystyle\sum_{i=1}^n x_i=1$. The answer is $x_i=\displaystyle\frac{w_i}{\sum_i w_i}$ but I don't know why apart from ...
2
votes
3answers
49 views

Triple finite sum

$\displaystyle \sum_{i=1}^a \sum_{j=1}^b \sum_{k=1}^c f(i,j,k)$ where a,b,c are fixed natural numbers and assuming $f(i,j,k)=i+j+k$. How do we calculate that sum? I mean is there any type for that ...
1
vote
2answers
183 views

Is $1 + 2 + 3 + \dots = -\frac{1}{12}$ really true? [duplicate]

I've read this strange result of the sum of all positive integers being $-\frac{1}{12}$. Is it really true? Does this also mean this is true? $$\sum_{n=1}^k n = \frac{k\cdot(k+1)}{2}$$ ...
0
votes
3answers
35 views

Lagrange Identity Proof

Was reading through Lagrange Identity Proof. However, one thing the proof assumes is $$\sum_{i=1}^p\sum_{j=1}^q a_i b_j=\sum_{i=1}^pa_i\sum_{j=1}^qb_j$$ which seems intuitive - but I wonder if ...