5
votes
3answers
660 views

Preventing “proof by homework”?

I am doing problem 3d in the Prologue of Spivak: Prove $(a+b)^n = a^n + {n\choose1}a^{n-1}b + {n\choose2}a^{n-2}b^2 + ... + {n\choose n-1}ab^{n-1} + b^n$ I feel like my proof could pass off as ...
1
vote
2answers
43 views

Bounding $\sum_{n=n_1}^\infty x^n (n+1)^2$

I need to upperbound the sum $$\sum_{n=n_1}^\infty x^n (n+1)^2$$ where $0<x<1$ is a parameter. I know it can be done starting from $$\sum_{n=n_1}^\infty x^n (n+1)^2\le \sum_{n=0}^\infty x^n ...
0
votes
0answers
27 views

How does one change the top number in a summation?

Sorry I do not know the correct term (I am guessing "upper limit"). Here is what I mean. $$\sum\limits_{i=1}^{\color{red}{17}}\frac{2i}{i+3}$$ The $17$ is what I am talking about as "the top number". ...
1
vote
3answers
44 views

Evaluating $\sum_{k=1}^{30} k(30-k)$

I tried to rewrite it as $\sum_{k=1}^{30} k(30-\sum_{k=1}^{30}k)$ and then replace the $\sum_{k=1}^{30} k$ with $\frac{n(n+1)}{2}$ then substitute $n=30$ into the equation, however I am not getting ...
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 ...
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
3
votes
2answers
68 views

Simplify a triple sum

I need to find a closed form for this summation: $$\sum_{j=1}^m\sum_{i=j}^m\sum_{k=j}^m\frac{{m\choose i}{{m}\choose{k}}}{j{m\choose j}}r^{k-j+i}$$ I posted this a long time ago, but today I found out ...
3
votes
1answer
149 views

Sum this series $\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\ldots$ upto $n$ terms

Sum this series: $$\dfrac{1}{1+1^2+1^4}+\dfrac{2}{1+2^2+2^4}+\ldots$$ upto $n$ terms. My approach: $$(1-n^6)=(1-n^2)(1+n^2+n^4)\implies \dfrac{n}{1+n^2+n^4}=\dfrac{n(1-n^2)}{1-n^6}$$ So, the ...
0
votes
1answer
39 views

Calculating a Summation Involving 2 Variables

Having not taken a math course for multiple years, I appear to have forgotten some bare basics. Unfortunately, Google has not taken me to a solid answer after much searching. How do you solve an ...
0
votes
1answer
96 views

$2+3+5+9+8+15+11+21+14+27+17+\dots$

I need to find the sum of the series upto $2n+1$ term $2+3+5+9+8+15+11+21+14+27+17+\dots$ so $2,5,8,11,14,17\dots$ is one AP and $3,9,15\dots$ is another AP I just want to know From which AP I ...
1
vote
0answers
63 views

Is there a general product formula for $\sum\limits_{k=1}^{n} k^p$

I'm familiar with Faulhaber's formula to express this sum as a much simpler one, but it appears that for any $p$ there's a product formula in $n$ for the sum e.g.: $$\begin{align} & ...
0
votes
0answers
25 views

$\sum_{i=1}^{a}\sum_{j=1}^{b}\sum_{m=1}^{n_{ij}}(\bar y_{i..}-\bar y_{…})\times (\bar y_{.j.}-\bar y_{…})=0$

I have to show that $$\sum_{i=1}^{a}\sum_{j=1}^{b}\sum_{m=1}^{n_{ij}}(\bar y_{i..}-\bar y_{...})\times (\bar y_{.j.}-\bar y_{...})=0$$ where $\bar ...
1
vote
2answers
49 views

How can I come up with a formula for this summation?

I have to come up with a formula for: $$\sum_{0\le i\le n\text{, i is even}}^\ i^2$$ and then prove it by using induction. I know how to do the proof, but I am stuck on coming up with the formula. I ...
1
vote
2answers
78 views

I'm stuck with this summation problem

How can I show that $$\displaystyle\sum_{n=1}^\infty \left(-\frac{1}{2^n}\left(\bigg| {\rm sgn} \left( {\rm round}\ (n \cdot \frac{1}{3})-n \cdot \frac{1}{3}\right)\right)-1\bigg|\right) = ...
0
votes
1answer
23 views

Bounding the sum of “almost” factorials

I am analyzing the complexity of an algorithm and the result is the sum of n products. Product 1 is the factorial. Product 2 is the factorial divided by 2. Product 3 is the factorial divided by 3 etc. ...
3
votes
2answers
71 views

$2^n=C_0+C_1+\dots+C_n$

Could anyone give me hints for this one? $2^n=C_0+C_1+\dots+C_n$ Thats all I can say and I know tricks like integrating or differentiating both side and then put $x=1$ or what ever we need. but ...
2
votes
2answers
150 views

Use induction and Newton's binomial formula to show that $\binom{n}{0}+\binom{n}{1}+\cdot+\binom{n}{n}=2^n, \forall n\in \mathbb N$ [duplicate]

Use induction and Newton's binomial formula to show that: $ i)$ $ \binom{n}{0}+\binom{n}{1}+\cdots+\binom{n}{n}=2^n, \forall n\in \mathbb N$ $ ii)$ ...
2
votes
0answers
174 views

How to calculate this triple summation?

I need to calculate the following summation: $$\sum_{j=1}^m\sum_{i=j}^m\sum_{k=j}^m\frac{{m\choose i}{{m-j}\choose{k-j}}}{k\choose j}r^{k-j+i}$$ I do not know if it is a well-known summation or not. ...
11
votes
10answers
2k views

Find five consecutive odd integers such that their sum is $55$.

So my professor asked us to do an Olympiad exercise which says that the sum of five consecutive odd integers is $55$, find those integers. But I've never seen such an exercise so it is quite new and ...
3
votes
1answer
71 views

How to calculate this summation?

I need to calculate this summation: $$\sum_{n=1}^\infty \frac{a^n}{n\cdot n!}$$ I know the answer without the excess $n$ in the denominator, i.e. $\displaystyle\sum_{n=1}^\infty ...
5
votes
2answers
111 views

Sequential sums $1+2+\cdots+N$ that are squares [duplicate]

While playing with sums $S_n = 1+\cdots+n$ of integers, I have just come across some "mathematical magic" I have no explanation and no proof for. Maybe you can give me some comments on this: I had ...
0
votes
4answers
79 views

Arithmetic summation question - I don't understand this answer.

I'm really sorry for this basic, stupid question. I have been looking for answers online but I can't find any. I don't understand the following summation: $$\sum_{i=0}^{n-1}i=\frac{n(n-1)}2$$ I ...
0
votes
2answers
30 views

Is this possible?

Is it possible to convert this forumla $$r(\theta) = \sum_{n=0}^\theta\left(\frac{2n+1}{2}\right)$$ to one without the $\sum$ sign? If so, how?
0
votes
2answers
87 views

How can I compute this sum of binomial

Is there any way to compute the following sum: $\displaystyle{ \sum_{\ell = {n + 1 \over{\vphantom{\LARGE A}2}}}^{n}{n \choose \ell}5^{n - \ell}}$ where $n$ is odd. Thank you.
0
votes
0answers
54 views

Close formula for triple sum binomial coefficient

I need to compute the following sum or to find a lower and upper bound that limit the sum: $\sum_{l=\frac{n+1}{2}}^n \binom{n}{l} \sum_{t=0}^{n-l} \binom{l}{t} 2^{l-t} \sum_{m=t}^{n-l} \binom{n-l}{m} ...
0
votes
1answer
85 views

Simplifying Sigma Notation

I am working on the proof on showing the ratio of two consecutive Fibonacci numbers converges to the golden ratio to explain to a student I am tutoring. I am getting to some confusion in a ...
0
votes
1answer
27 views

How can i simplify the following term to get the right side?

$$\sum_{h=1}^{L}\frac{W_h^2S_h^2}{n_h}=\frac{1}{n}\sum_{h=1}^{L}{(W_hS_h)}^2$$ where, $n_h=\frac{n}{\sum_{h=1}^{L}N_hS_h}N_hS_h$ $\quad\text{and}\quad$ $W_h=\frac{N_h}{N}$ $\quad\text{and}\quad$ ...
0
votes
1answer
42 views

Solution of a series equation containing $\pi$

I have to find the coefficients $a_k$ for which the following equation is satisfied: $$S(N)=\sum_{k=1}^Na_k\frac{\pi^{k+3}}{k+3}=-2\pi+\epsilon(N)$$ where $\epsilon(N)$ is an error depending on $N$. I ...
3
votes
2answers
189 views

Sum of derangements and binomial coefficients

I'm trying to find the closed form for the following formula $$\sum_{i=0}^n {n \choose i} D(i)$$ where $D(i)$ is the number of derangement for $i$ elements. A derangement is a permutation in which ...
0
votes
2answers
42 views

Simplifying Multiple Summations for worst case analysis

I'm figuring out a worst case analysis on a function. After converting it to a set of summations, and changing the sigma notations into summation formuale I ended up with: ...
0
votes
0answers
30 views

Quick simplification strategy for $\binom{3}{2}p^2(1-p) \le \sum_{k=3}^{5}\binom{5}{k}p^k(1-p)^{n-k}$

What is a quick simplification strategy to solve the following expression for $p$ by hand? (or less preferably, by a TI83/86 calculator). $$\binom{3}{2}p^2(1-p) \le ...
1
vote
0answers
28 views

Are there efficient ways of computing sums that involve trigonometric functions and q-logarithms (Tsallis q-logarithms)?

I am interested in computing the following sum: \begin{equation} \sum\limits_{l=1}^k l^{\beta_1} \cos\left(\omega \log_q(\frac{l}{t_c})\right) \end{equation} Here $0 < \omega$, $0 < k < ...
2
votes
1answer
70 views

Sum of series ${n\choose 2a}{a\choose 0}+ {n\choose {2a+2}}{{a+1}\choose 1} + {n\choose {2a+4}}{{a+2}\choose 2} + \ldots$

I wanted to check the rationality of the cosine function for some rational multiples of $\pi$. And I found out that, $\cos(n \cdot\arccos x)$ generates a polynomial in $x$ whose co-efficients have the ...
0
votes
1answer
56 views

How to read the notations in the proof of the inclusion-exclusion principle?

This question is a follow up to this question. How do I create a sequence from this formula? $\sum\limits_{i=1}^{n} P(A_i) - \sum\limits_{1\le i < j \le n-1}P(A_i \cap A_j) + \dots + (-1)^n ...
7
votes
1answer
94 views

Find the sum of the first ten terms

How do I find the sum below? $$\sum_{i=1}^{10}\frac{2i+1}{i^2(i+1)^2}$$ I think there should be a simpler way instead of just adding the ten terms up using brute force, since it's impossible to do ...
2
votes
1answer
68 views

Sum of squares of power of two sums and differences

What is the sum $$\sum\left(\frac12-\frac1{2^n}\pm\frac14\pm\frac18\pm\ldots\pm\frac1{2^n}\right)^2$$ where the sum is taken over all $2^{n-1}$ combinations of plus and minus signs? If the sum is too ...
1
vote
2answers
41 views

Variance of Portfolio Return

I am struggling to understand the following proof. I do not understand the highlighted red step. Please could this be broken down so that I can understand it. It seems like a manipulation of ...
0
votes
0answers
123 views

factorial moments of hypergeometric distribution

Factorial moment of positive order : $$\mu_k=\mathbb E[X(X-1)\ldots(X-k+1)]$$ $$=\sum_{m=0}^{n}m(m-1)\ldots(m-k+1)\frac{\binom{a}{m}\binom{b}{n-m}}{\binom{a+b}{n}}$$ ...
0
votes
1answer
122 views

How to find a compact expression for $\sum\limits_{k=1}^n \frac k{(K+1)!}$?

$$\sum_{k=1}^n \frac k{(K+1)!}$$ How to find a compact expression? (Original scan here)
0
votes
1answer
33 views

Arrangement of the following term

How $$\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{2m}}$$ can be rewritten as $$\sum_{k=1}^{\infty}\frac{(-1)^{2k-2}}{(2k-1)^{2m}}+\sum_{k=1}^{\infty}\frac{(-1)^{2k-1}}{(2k)^{2m}}$$ ???
2
votes
1answer
72 views

Pulling sigma notation out of thin air

The function $ f(t) $ satisfies $ \, 0 \leq f(t) \leq k $ for $ \, 0 \leq t \leq 1 $, where $ k $ is a constant. Therefore, $$ 0 \leq \int_{0}^{1}f(t)\, dt \leq k $$ So, if $$ f(t) = \frac{1}{n-t}\, ...
1
vote
1answer
62 views

Summation of series

Write down the sum of $\displaystyle \sum_1^{2N} n^3$ in terms of $N$, and hence find: $1^3 - 2^3 + 3^3 - 4^3 + \cdots - (2N)^3$ in terms of $N$, simplifying your answer. I found this to be ...
0
votes
2answers
73 views

Is it possible to evaluate $\sum_{r=1}^{20}\frac{1}{r(r+1)}$ using $\sum_{r=1}^{n}\frac12n(n+1)$ [duplicate]

Evaluate $$\sum_{r=1}^{20}\frac{1}{r(r+1)}$$ It splits into $$\sum_{r=1}^{20}\frac{1}{r}-\sum_{r=1}^{20}\frac{1}{r+1}$$ I'm stuck on how to apply the standard result $\sum_{r=1}^{n}\frac12n(n+1)$ to ...
2
votes
2answers
250 views

Closed-form Geometric series of both increasing and decreasing variables?

This question comes from the formula $$x^n - a^n = (x-a)(x^{n-1}a^0 + x^{n-2}a^1 + .... + x^1a^{n-2} + x^0a^{n-1})$$ which can be verified by summing the second factor as a geometric series. My ...
1
vote
1answer
48 views

Is there a way to change the order of these summation terms

$$\sum_{K=2}^{N}\sum_{L=1}^{\lfloor\frac{K}{2}\rfloor-1}$$ I want to have the $L$ summation on the outside and the $K$ summation on the inside somehow. Can this be done?
5
votes
3answers
208 views

How to find $\sum n^3$ if $\sum n^2$ is given

Problem : Find $\sum n^3$ if $\sum n^2 =2870$ Can we use the following method : $\sum n^2 = \frac{n(n+1)(2n+1)}{6}$ = 2870.. ( As sum of the square of first n natural number is ...
3
votes
1answer
88 views

Square sum vs. Sum square

Is there any condition that make them as equal? $$ \sum_{i=0}^n (x_i^2+y_i^2) = \left( \sum_{i=0}^n x_i \right)^2 + \left( \sum_{i=0}^n y_i \right)^2.$$ I think above two equations have difference ...
5
votes
2answers
558 views

How to find the sum of this series : $1+\frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}$

Problem : How to find the sum of this series : $1+\frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}$ This is a Harmonic progression : So is this formula correct to sum the series : ...
0
votes
0answers
103 views

Simplify the math expression $M=(1-b)^{N}+(N-4)(1-b)^{N-1} b+ \sum^{N}_{i=1} 2^{2i-2} \Delta^2 (1-b)^{N-1}b-(1-2b)^2$

Can someone help me to further simplify the following expression? Here, $0<b<1$ and we can assume that $b$ is small. $\Delta$ is a constant. Thank you $M=(1-b)^{N}+(N-4)(1-b)^{N-1} b+ ...
1
vote
1answer
50 views

Simplify Mathematical Expression

Can someone help me to simplify the following expression? I can assume b is small and $0<b<1$. $(C^{N}_{i})$ is the binomial coefficient. $$A=[(1-b)^{N} + \sum^{N}_{i=1} (C^{N}_{i} - 2 ...