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

learn more… | top users | synonyms

0
votes
0answers
8 views

How can you find the integer part of y starting from this inequality?

How can you find the integer part of y starting from this equality? (I need a precise procedure, not only the number)
0
votes
1answer
20 views

Partial sum formula of a polynomial series?

I am trying to find the partial sum formula of the following series: $$ \sum_{y=1}^{\infty} \frac{4y^2-12y+9}{(y+3)(y+2)(y+1)y} $$ I have tried using Faulhaber's formula without success. I have also ...
0
votes
2answers
26 views

Finding a sum of alternating cubes

Find this sum, in terms of $n$: I have some hand-written hints from someone else, but can't read his writing:
3
votes
2answers
80 views

Estimating sum with binomial coefficients

Lately when I was estimating complexity of some algorithm I came across this sum: $$\sum_{k=0}^n \binom {n}{k} \binom {n-k}{k}$$ Is it possible to find a closed-form expression for this sum, or at ...
2
votes
1answer
49 views

Find the closed form of an n-sum

I'd like to find the closed form or a quickly converging rewriting of the following n-sum: ...
1
vote
1answer
31 views

Infinite sum question

How to solve the following infinite sum: $$\sum_{n=0}^\infty \frac{[(-25)^n+(-9)^n+(-1)^n]x^{(2n)}}{(2n)!}$$ I have no idea where to go from here, but I began with $\cos(5x) + \cos(3x) + \cos(x)$. I ...
1
vote
0answers
36 views

Give a combinatorial proof [duplicate]

$$\sum_{k=1}^n k{n\choose k}=n\cdot 2^{n-1}$$ I have to prove the identity using a combinatorial proof: I think this should be my combinatorial proof. We want to form a committee of $k$ people from ...
7
votes
1answer
110 views

A formula for $\lfloor n\rfloor+\left\lfloor \frac n2\right\rfloor+ \left\lfloor \frac n3\right\rfloor+\ldots+\left\lfloor \frac nk\right\rfloor$?

Is there any formula to calculate: $$\lfloor n\rfloor+\left\lfloor \frac n2\right\rfloor+ \left\lfloor \frac n3\right\rfloor+\ldots+\left\lfloor \frac nk\right\rfloor$$ with $n$ and $k$ positive ...
2
votes
2answers
30 views

Another Summation

After looking at the question here Computing summation I wondered if it might be possible to evaluate the following summation with a similar-looking summand term but with $2n$ instead of $2^n$: ...
0
votes
0answers
14 views

Minimizing sum of products

Consider a total of $d$ items, $\{I_1, I_2, \cdots, I_d \}$, each having a weight $w_i$, and a total of $m$ bins, $\{B_1, B_2, \cdots, B_m\}$. We would like to distribute the items into the bins such ...
0
votes
1answer
10 views

Showing the Summation of $(\frac{w}{2})^k$ where w is a complex root

I got the correct answer for (i) and (ii) and the problem is with third part. I cant find my mistake. Since the third part is related to the second part, I will mention its answer. The ...
0
votes
1answer
34 views

Evaluating a sum by applying geometry

This is really an interesting question: Evaluate S, where $$ \large S= \sum_{k=1}^{502} \left\lfloor \frac{305k}{503}\right\rfloor$$
14
votes
1answer
118 views

Trig sum: $\tan ^21^\circ+\tan ^22^\circ+…+\tan^2 89^\circ = ?$

As the title suggests, I'm trying to find the sum $$\tan^21^\circ+\tan^2 2^\circ+...+\tan^2 89^\circ$$ I'm looking for a solution that doesn't involve complex numbers, or any other advanced branch in ...
0
votes
1answer
32 views

combinatorial proof of summation

Prove $\sum_{i=1}^n2^{i-1}=\sum_{i=0}^{n-1}2^i=2^n-1$ combinatorially. This is easy to prove inductively. I know that $\sum_{i=0}^n{n\choose i}=2^n$ so maybe change $\sum_{i=0}^{n-1}2^i$ to ...
-1
votes
2answers
49 views

What's the result of $\sum_{n=0}^{\infty} k^n/n!^2$? [on hold]

As the title says, I'd like to know the result of: $\sum_{n=0}^{\infty} \frac{k^n}{n!^2}$ Thank you very much! Edit: The story goes: two independent Poisson processes with success probabilities ...
8
votes
3answers
108 views

Is it true that $\left\lfloor\sum_{s=1}^n\operatorname{Li}_s\left(\frac 1k \right)\right\rfloor\stackrel{?}{=}\left\lfloor\frac nk \right\rfloor$

While studying polylogarithms I observed the following. Let $n>0$ and $k>1$ be integers. Is the following statement true? $$\left\lfloor \sum_{s=1}^n \operatorname{Li}_s\left( \frac{1}{k} ...
0
votes
1answer
41 views

How to deal with summation with log bounds like: $\sum \limits_{i=1}^{\lg (n)}$ [on hold]

I came across this summation in my algorithms textbook. I've googled everywhere and can't seem to find anything on how to deal with these types of bounds. (Apparently it equals 2n as n approaches ...
1
vote
0answers
40 views

What is the largest value one can get in game 2048 without new tiles appear

This is a simplified version of the famous game 2048. Given a 4x4 grids with some values chosen from {0, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048}. A value of 0 indicates that the position in ...
3
votes
2answers
22 views

Why is $\displaystyle\sum_{k=j}^{i+j}(j+i-k) = \displaystyle\sum_{k=1}^{i}(k)$

$\displaystyle\sum_{k=j}^{i+j}(j+i-k) = \displaystyle\sum_{k=1}^{i}(k)$ I know the above are equal through testing it out with arbitrary values, but I can't get an intuitive grasp as to why this is. ...
0
votes
0answers
17 views

differentiation of log of a sum of vectors

I would very much like to be able to differentiate the following function with respect to $x$: $$ln(\sum \limits_{j=1}^{d} \theta_j \bf a_j^2)$$ Where $\bf a_j$ is a $d \times 1$ vector with $x$ in ...
0
votes
0answers
19 views

Turning every possible summation into integral form with this formula?

I discovered a formula for fun, that turns a summation into an integral: $$\sum_{x=a}^{b} f(x) = \int_{a}^{b+1}f\left(x-0.5-\frac{arctan(tan(\pi*(x-0.5)))}{\pi}\right) dx $$ This is probably useless, ...
1
vote
3answers
45 views

Yet another sum involving binomial coefficients.

Given $A,B,N \in \mathbb N$ Is there a closed form for this expression? $$\sum_{n=1}^N n \binom{A}n \binom{B}{N-n} $$ If there is such, can you give a proof? EDIT: $A,B \geq N$
0
votes
4answers
37 views

summation algebra for $\sum_{n=0}^\infty x^n + \sum_{n=0}^\infty x^{n+1}$

Why does $\sum_{n=0}^\infty x^n + \sum_{n=0}^\infty x^{n+1} = 1 + 2\sum_{n=1}^\infty x^n$? Shouldn't this be $1 + x + 2\sum_{n=1}^\infty x^n$ because of the $n+1$ in the second summation?
1
vote
2answers
62 views

Prove $1(1!)+\dots+n(n!) = (n+1)!-1$ using induction

So I'm trying to prove this statement (through induction): $$1(1!)+2(2!)+\dots +n(n!)=(n+1)!-1$$ But I'm confused with the inductive step here: $$(n+1)!-1+[(n+1)(n+1)!] = (n+2)!-1$$ What do I do ...
3
votes
6answers
153 views

Evaluating $\lim_{n\to\infty} \frac{1^{99} + 2^{99} + \cdots + n^{99}}{n^{100}}$ using integral

Evaluate $\lim\limits_{n\to\infty} \dfrac{1^{99} + 2^{99} + \cdots + n^{99}}{n^{100}}$ This is the question I remembered from my high school textbook (I remembered it while reading about ...
2
votes
2answers
452 views

Sum of real numbers that multiply to 1

I've seen a question in my math book with this explanation above it: "If the product of n positive real numbers is 1, then the sum of these numbers must be more than n". I was wondering if this is ...
1
vote
0answers
18 views

Approximating the max value of a function containing the half-sum of binomial series

I recently met a problem and I have been finding related materials about it. However it is still hard to handle. Precisely, I want to maximize a function related to $$S_n = \sum_{i=0}^{\lfloor ...
2
votes
1answer
26 views

Saving for retirement - how much?

I'm working through a problem in the book "An Undergraduate Introduction to Financial Mathematics" and there is an example I can follow. The problem is: Suppose you want to save for retirement. The ...
2
votes
1answer
37 views

What is the infinite sum of logarithm lnK divided by k(K+1)?

I was trying to calculate the following integral: $\displaystyle\int_1^\infty\frac{dx}{x \lfloor x \rfloor}=? $ which I found to be equivalent to $\displaystyle\sum_{k=2}^\infty\frac{\ln(k)}{k(k+1)} ...
0
votes
0answers
29 views

How fast can we approximate the sum of the tangent?

So Wikipedia gives the sum of the tangent on this page as: \begin{align} \sum_x{\tan{(x)}} &= ix - \psi_{e^{2i}}(x + \pi/2)+C \\ &= -\sum_{k=1}^\infty{ \psi(k \pi - \pi/2 + 1 - x)} \\ &- ...
0
votes
1answer
39 views

Riemann Zeta of 1/2 $\zeta(\frac{1}{2})$

This may be a silly question, but I need to figure out how to evaluate the value of $\zeta(\frac{1}{2})$. In wikipedia, it says: $\zeta(1/2) \approx -1.4603545$. I am interested to know how this value ...
3
votes
5answers
128 views

Proving the summation formula using induction: $\sum_{k=1}^n \frac{1}{k(k+1)} = 1-\frac{1}{n+1}$

I am trying to prove the summation formula using induction: $$\sum_{k=1}^n \frac{1}{k(k+1)} = 1-\frac{1}{n+1}$$ So far I have... Base case: Let n=1 and test $\frac{1}{k(k+1)} = 1-\frac{1}{n+1}$ ...
0
votes
2answers
27 views

Simplifying a sigma expression with square roots in the denominator?

Let $$A = \frac{1}{\sqrt{n}\sqrt{n + 1}} + \frac{1}{\sqrt{n}\sqrt{n + 2}}+...+\frac{1}{\sqrt{n}\sqrt{n + n}}$$ where $n$ is any real number such that, $$A = \sum_{i = ...
0
votes
1answer
48 views

Prove this inequality by math induction

$$\sum \limits_{k=1}^{n-1} k^p < \frac{ n^{p+1}}{p+1} < \sum\limits_{k=1}^n k^p $$ I know how to prove it by using Riemann Sum, but it I was thinking if there is anyway to do it by mathematical ...
1
vote
4answers
39 views

How to find closed form of summation of Fibonacci Sequence?

I created two formulas to prove a binary theory involving the Fibonacci sequence. (1) $\sum_{i=0}^n F_{2i+1} $ Equation (1) is the sum of all Fibonacci numbers up to $F_n$ where every $i$ in $F_i$ ...
0
votes
1answer
17 views

differentiating a summation series

How would one go about partially differentiating the following with respect to L; z = $\frac{T}{L}\Psi_{z} - \frac{T}{L}\sum_{n=1}^N X(n) sinh[\frac{2 \pi n}{L}(h + z)] cos(\frac{2 \pi n}{L}x)$ I ...
5
votes
0answers
67 views

Double factorial as a sum

I believe the following equality to hold for all integer $l\geq 1$ $$(2l+1)!2^l\sum_{k=0}^l\frac{(-1)^k(l-k)!}{k!(2l-2k+1)!4^k}=(-1)^l(2l-1)!!$$ (it's correct for at least $l=1,2,3,4$), but cannot ...
0
votes
0answers
19 views

Combinatorics Summation Simplification [duplicate]

I am at a loss for how to approach simplifying the following combinatorics summation. I assume some method using summation formulas and combinatorial identities is required. $$\sum_{c = ...
2
votes
1answer
50 views

Using the Fourier Series of $f(t)=(t-\frac{1}{2})^{2}$ to deduce the sum $\sum_{n=1}^{\infty }\frac{1}{n^{2}}$?

So this is a question in one of the previous tests: My approach (if you want just skip to step 3.):$$$$ 1. Formulation of the problem and calculating the constant term of the series $a_o$ I ...
0
votes
1answer
53 views

Conditions for equality of two binomial sums

Let $k,r,n$ be integers such that $0<k,r<n$. Let $$K=\sum^n_{i=k}k^{n-i}\binom{n-k}{i-k}^2k!(i-k)! \,\text{ and }\, R=\sum^n_{i=r}r^{n-i}\binom{n-r}{i-r}^2r!(i-r)!.$$ How to show that ...
0
votes
5answers
48 views

Combinatorial argument for the sum of the first $n$ integers.

Can someone give a combinatorial argument (at least for $\binom{n+1}{2}$) for why $\binom{n+1}{2}=(n^2+n)/2$?
0
votes
3answers
40 views

Solving this summation: $\sum_{i=1}^k i\cdot2^i$ [duplicate]

$$\sum_{i=1}^k i\cdot2^i$$ I'm working on a recurrence relation question and now I'm stuck at this point. I have no idea how to simplify this down to something I can work with. Can I seperate the ...
0
votes
1answer
66 views

Infinitely Many Circles in an Equilateral Triangle

In the figure there are infinitely many circles approaching the vertices of an equilateral triangle, each circle touching other circles and sides of the triangle. If the triangle has sides of length ...
0
votes
0answers
23 views

Exchanging the limit of an integral with a finite sum

So, in general, I can get this value: $$\lim_{a \to \text{a constant}}{ \int{ \left( \sum_{x=x_1}^{x_2}{ f(a,x) }da \right)} } \tag{1}$$ What I'm after is this: $$\sum_{x=x_1}^{x_2}{\left( \lim_{a ...
1
vote
3answers
74 views

How to evaluate the following? $\sum\limits_{n=1}^{\infty} \int_1^{n!} x^{-n}{\rm d}x$.

How can I evaluate this sum? $$\sum_{n=1}^{\infty} \int_1^{n!} x^{-n} \mathrm{d} x$$ Is there a closed form or a transform that would make it possible not to "CAS" it? Also, it seems to converge, but ...
1
vote
3answers
32 views

Find $\sum{(-1)^22n }$ from $n=0$ to $n=28$

Find $\sum{(-1)^22n }$ from $n=0$ to $n=28$ I can't find a formula for alternation series
3
votes
1answer
94 views

Powers of 2 in the product of the Fibonacci numbers

I'm working on finding a summation that pulls the powers of 2 out of the product of the Fibonacci numbers. I've noticed some patterns for the Fibonacci number. For example. Looking at the Fibonacci ...
0
votes
3answers
23 views

Quotient of infinite products

I have a very simple question, but I never learned about infinite products and now have to use them. Am I right in assuming that $$ {\prod_{k=1}^{\infty} f(k)\over\prod_{k=1}^{\infty} ...
2
votes
5answers
96 views

How to compute $\sum_{k =1}^{100}(-1)^k$

Today I tried to compute $$ \sum_{k =1}^{100}(-1)^k $$ Is there a way to find the result more quickly ? Below if my attempt to find the result. Especially without considering the case of odd and ...
0
votes
2answers
52 views

Seeking proof using mathematical induction

\begin{equation}a: \mathbb N ×\mathbb N \to \mathbb R \end{equation} where for all \begin{equation}x,y\in\mathbb N\end{equation}\begin{equation}a(x,y) =a(y,x)\end{equation} How do I show that the ...