0
votes
2answers
75 views

The sum of $1^2+7^2+13^2+\cdots+n^2,$ where $ n =1 \mod6 $

Find the sum of this progression in the terms of $n$ $$1^2+7^2+13^2+\cdots+n^2,$$ where $ n =1 \mod6 $. Are Bernoulli's numbers involved in this? Please help.
1
vote
0answers
104 views

Evaluate this product $n \times \frac{n-1}{2} \times \dots \times \frac{n-(2^k-1)}{2^k}$

For $k = \lfloor \log_{2}(n+1) \rfloor - 1$ evaluate $n \times \frac{n-1}{2} \times\frac{n-3}{4} \times \frac{n-7}{8} \times \dots \times \frac{n-(2^{k}-1)}{2^k}$ So the product goes up to $k$ and I ...
4
votes
2answers
344 views

n more each day

It's been a while since I've been at school and I don't work in a field that practices this sort of stuff so I don't know the formula my brain can't wrap my head around the problem. The problem: You ...
-1
votes
1answer
65 views

Rewrite and approximate the sum as an Integral $\sum_{i=1}^{1000} \sqrt{i}$ [closed]

This is not an Infinite sum !, how do we change this to an Integral. $ $ We normally write an integral as an infinite sum.
1
vote
1answer
116 views

Find the sum of the series $\sum \limits_{n=3}^{\infty} \dfrac{1}{n^5-5n^3+4n}$

Feel free to skip obvious steps, or use a calculator when required. I just want to understand the theme of the solution. Any help is appreciated EDIT : We can write$$ \dfrac{1}{n^5-5n^3+4n} = ...
0
votes
2answers
31 views

If a sequence $\{a_n\}$ satisfies the Inequality $a_{n+1} < ka_{n}$, then show that $ \lim\limits_{n \to \infty} a_n =0$ where $0< k , a_n< 1$

I know one solution. Consider $\sum a_n$ Then use ratio test to show that the series converges, hence the sequence. Any other Ideass !
3
votes
1answer
52 views

Give example of a series $\sum a_n$ such that the series is conditionally convergent. and $\sum na_n$ is convergent

I tried all the conditionally convergent series I know, I found $\sum na_n$ to be diverging for all of them. But I am sure the question is correct
0
votes
5answers
74 views

Refresh summation formulas

I am trying to refresh on algorithm analysis. I am looking for a refresher on summation formulas. E.g. I can derive the $$\sum_{i = 0}^{N-1}i$$ to be N(N-1)/2 but I am rusty on the and more complex ...
4
votes
2answers
97 views

Evaluating $\sum_{n=1}^{99}\sin(n)$ [duplicate]

I'm looking for a trick, or a quick way to evaluate the sum $\displaystyle{\sum_{n=1}^{99}\sin(n)}$. I was thinking of applying a sum to product formula, but that doesn't seem to help the situation. ...
2
votes
5answers
108 views

Finding the minimum value of a sum [closed]

Let $x,y,z$ be real numbers . Find the real number $a$ so that $S$ has a minimum value , where $$S=|x-a|+|y-a|+|z-a| .$$
4
votes
1answer
83 views

An inverse binomial summation.

I am looking for a closed form for this summation: $$ \sum_{j=1}^m\frac{r^{-j}}{j{m\choose j}} = \sum_{j=1}^m\frac{r^{-j}}{m{m-1\choose j-1}} = \frac1{rm} \sum_{k=0}^{m-1}\frac{r^{-k}}{{m-1\choose k}} ...
1
vote
6answers
61 views

Why does $ 1+2+3+…+p = {(1⁄2)}\cdotp\cdot(p+1) $ [duplicate]

I saw this from Project Euler, problem #1: If we now also note that $ 1+2+3+...+p = {(1/2)} \cdot p\cdot(p+1) $ What is the intuitive explanation for this? How would I go about deriving the latter ...
2
votes
3answers
42 views

Expression generating $\left( \frac{3}{10}, \, \frac{3}{10} + \frac{33}{100}, \, \frac{3}{10} + \frac{33}{100} + \frac{333}{1000}, \dots \right)$

I'm looking for a closed-form expression (in terms of $n$), that will give the sequence $$ (s_n) = \left( \frac{3}{10}, \, \frac{3}{10} + \frac{33}{100}, \, \frac{3}{10} + \frac{33}{100} + ...
5
votes
3answers
695 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
29 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
46 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
18 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
84 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
1answer
56 views

On $\lfloor\sqrt n \rfloor+ \sum_{j=1}^n \lfloor n/j\rfloor$

How do we prove that $\Big[\sqrt n \Big]+ \sum_{j=1}^n \bigg[ \dfrac nj\bigg]$ is an even integer for all $ n \in \mathbb N$ ? (where $\Big[ \space \Big]$ denotes the "greatest integer" function)
3
votes
2answers
83 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
159 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
50 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
75 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
26 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
51 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
86 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
25 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
72 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
175 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
223 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
73 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
114 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
84 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
33 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
88 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
65 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
95 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
28 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
45 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
223 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
48 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
29 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
73 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
59 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
98 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
78 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 ...