0
votes
2answers
40 views

Calculate the following sequence $\sum_{n=0}^{+\infty }\left ( -\dfrac{1}{4\alpha } \right )^{n}\dfrac{ (2n)!}{n!},\; \alpha >0$

Calculate the following sequence $$\sum_{n=0}^{+\infty }\left ( -\dfrac{1}{4\alpha } \right )^{n}\dfrac{ (2n)!}{n!},\; \alpha >0$$
4
votes
1answer
76 views

How can I prove that $\frac{\sigma(n)}{n} = \sum_{(d|n)} \frac{1}{d}$ for every $n \in \mathbb{Z^{+}}$?

I want to show that $\displaystyle \frac{\sigma(n)}{n} = \sum_{(d|n)} \frac{1}{d}$ for every $n \in \mathbb{Z^{+}}$. This is essentially a basic number theory question. I am able to get to the ...
1
vote
1answer
39 views

What can I do to this expression to lose the summations?

I'm at the end of a past paper question and need to derive this answer: I am very close and have got to this by doing d/dx to the * equation: What can I do to get rid of these summation signs ...
0
votes
0answers
20 views

Summation of a function with the variable both in the function amd in the upper limit

E is defined as : E = c1 ( a$\rho$ + b$\rho ^{2}$ ) + c2 $\rho$ ( c + d $\sum_{j=0}^{n} (\log{ \frac{R\rho}{j} } ) $ ) + c3 $\rho ^{2}$ a, b, c, d, c1, c2, c3, R are known constants. $\rho$ is the ...
0
votes
1answer
22 views

Determining value of infinite sum after computing full Fourier Series

I have computed the Full Fourier Series of the function $\phi:[-\pi,\pi] \rightarrow \Bbb{R}$ defined by $\forall x \epsilon[-\pi,\pi], \phi(x)=|\sin(x)|$ to be: $$ \phi(x) = {2\over\pi}+{1\over\pi} ...
2
votes
4answers
91 views

Exact value of $\sum\limits_{n=1}^\infty(-1)^{n(n+1)/2}/n$?

Wolfram is not computing it properly. What is the exact value of $$\sum_{n=1}^\infty\frac{(-1)^{n(n+1)/2}}{n}?$$ How to avoid imaginary $i$ coming from the exponent?
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
2
votes
1answer
56 views

Exponentiation in terms of Summation

For positive integers, $a \times b=\sum\limits^{b}{a}$, correct? So therefore exponentiation where n is also a positive integer should be something like $a^n=\sum\limits^n{\sum\limits^a{a}}$ This is ...
3
votes
2answers
104 views

Showing that $\sum_{i=1}^n \frac{1}{i} \geq \log{n}$

I have been trying to prove this by induction on $n\in \mathbb{N}$, but this approach seemed to get me nowhere. I have a suspicion it might be necessary to express $\log{n}$ as $\int_1^n 1/x\text{ ...
0
votes
1answer
36 views

Inequality in non-decreasing sequence

Let $a, b$ be two sequences of real numbers such that $a_1 \le a_2 \le \dots \le a_n$ and $b_1 \le b_2 \le \dots \le b_n$. Prove (or disprove) that ...
0
votes
1answer
35 views

How to prove that convolution on real sequences is associative?

Given two real sequences $\{ a_n \}$ and $\{ b_n \}$, where $n \ge 0$, the convolution operation (denoted $\ast$) is defined as $\{ c_n \} = \{ a_n \} \ast \{ b_n \}$, where $c_n = \sum_{k=0}^{n} a_k ...
0
votes
1answer
66 views

Digit in units place of 1!+2!+…99!

There isn't much I can add to the question description to expand upon the title. I came across this in a multiple choice test. The options were 3, 0, 1 and 7. I am absolutely stumped. Any pointers? By ...
5
votes
2answers
269 views

Proving $\sum_{k=1}^n{2k-1\choose k}{2n-2k+1\choose n-k+1}=4^n-{2n+1\choose n+1}$

Some background. I was asked to find an arithmetic function $f$ such that $f*f=\mathbf 1$ where $\mathbf 1$ is the constant function 1 and $*$ denotes Dirichlet convolution. I was able to prove that ...
2
votes
0answers
77 views

How to prove this combinatorial identity

I am wondering how to prove the following identity: $$\sum_{k=0}^r {r-k \choose m} {s \choose k-t} (-1)^{k-t} = {r-t-s \choose r-t-m}$$ It seems that I can negating the upper index of ${s \choose k-t} ...
1
vote
4answers
85 views

Algebra-sum of entries in each column of a sqaure matrix = constant

This is a question from an algebra homework and I am just looking for some tips. The question is: We have: $M$: an $n\times n$ matrix with real entries $c$: a real constant the ...
0
votes
1answer
36 views

Difficulty with understanding summations [duplicate]

I am in advance sorry if this question is too easy for this site, but I am having real problem understanding how to solve this summation: $$\sum_{i=1}^n{i*2^i}$$ I understand basics of summations ...
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 ...
0
votes
1answer
71 views

Should be simple inductive proof

Establish the following recursion relations for means and variances. Let $\overline{X}_n$ and $S_n^2$ be the mean and variance, respectively, of $X_1,\dots,X_n$. Then suppose another observation, ...
1
vote
0answers
46 views

Summation of $2^{(-2^{n})}$ [duplicate]

By the ratio test, I know that this series convernges: $\sum2^{(-2^{n})}$, in the limit $n$ goes to infinity. Probably to something close to $.8$ (if not equal to $.8$). The problem is, how do I ...
0
votes
2answers
75 views

Find the symmetric matrix that represents the quadratic form $Q(X)=trace(X^2)$, $X\in mat_n\mathbb (R)$

as the title says, find the symmetric matrix (or signature) of $Q(X)=trace(X^2)$ where $X$ is an $n$ by $n$ matrix with real entries. the diagonal of $X^2$ is $$\sum_{k=1}^n x_{ik}x_{ki}$$ So ...
3
votes
1answer
94 views

Signature of quadratic form $Q(p)=p(1)p(2)+p(3)p(4)$

I was asked to find the signature of the quadtratic form $Q(p)=p(1)p(2)+p(3)p(4)$ where $p$ is a polynomial in $\mathbb R_n[x]$ I tried doing it via finding the symmetric matrix that $Q$ corresponds ...
0
votes
2answers
44 views

Recurrence relation - Show that a sum of a sequence is zero

We are given the following sequence: $f(n)=4f(n-1)-5f(n-2)$, $f(0)=f(1)=a$ where $a$ is some value in $\mathbb C$. We are asked to show that $$\sum_{n=0}^{\infty}\frac{f(n)}{3^n}=0$$ First thing I ...
2
votes
3answers
81 views

Solving recurrence relation with generating functions - Nearly got the answer

I'm trying to solve the following recurrence relation (Find closed formula) using generating functions: $f(n)=10f(n-1)-25f(n-2)$, $f(0)=0$, $f(1)=1$ I'm having a small difficulty at the end and can ...
1
vote
3answers
66 views

Finding limit of a product.

Prove:$$\lim_{n \to\infty }\frac{1}{n}\left[\prod_{i=1}^{n}(n+i) \right ]^{\frac{1}{n}}=\frac{4}{e}$$ I tried using Squeeze Theorem but can't go beyond $1<L<2$. $$\lim_{n\to\infty} \left( 1 + ...
3
votes
1answer
186 views

Find the sum of the series.

I need to find the following sum: $$\sum_{s=0}^{n+1}{(-1)}^{n-s}4^s\binom{n+s+1}{2s}$$ First I tried to simplify this: $$\begin{split} \sum_{s=0}^{n+1}{(-1)}^{n-s}4^s\binom{n+s+1}{2s} &= ...
2
votes
2answers
69 views

Help with compact notation for sum

I've already understood the motive of this sum using the nom-compact way, but I want to do it in the compact way, so it will be rigorous. Please, I need some help: ...
2
votes
3answers
80 views

Sum of $n$ terms of a series

Find the sum of $n$ terms of the following series: $$1+(1+x)+(1+x+x^2)+\cdots$$ The $n^{th}$ term $(t_n)$ is $\displaystyle\frac{x^n-1}{x-1}$, since each term is a Geometric Progression with common ...
5
votes
2answers
68 views

Proving a lower bound on the limit superior of a sequence.

Prove that for every positive sequence {$a_{n}$}, $$\varlimsup_{n \to \infty}\frac{\sum_{i=1}^{n+1}a_{i}}{a_{n}}\geq 4$$ Also find the sequences {$a_{n}$} for which 4 is attained. Attempted ...
3
votes
3answers
132 views

Find the 100th derivative of $x \sinh(2x)$

If $f(x) = x \sinh(2x)$, find $f^{({100})}(x)$. My (Incorrect) working so far: Using Leibniz' Formula for derivatives: $$(fg)^{(n)}=\sum_{k=0}^n{n\choose k}f^{(k)}g^{(n-k)}$$ ...
6
votes
3answers
85 views

Show that $\sum_{k=2012}^{n} 2^k\binom{n}{k} = \Theta(3^n)$

In this question we are asked to show that $\sum_{k=2012}^{n} 2^k\binom{n}{k} = \Theta(3^n)$ What I did: $\sum_{k=2012}^{n} 2^k\binom{n}{k} = \sum_{k=2012}^{n} 2^k*1^{n-k}\binom{n}{k} \leq ...
0
votes
3answers
64 views

Prove equation including sum of sequence and combinatorics

Prove that: $\sum\limits_{i=1}^n \frac{i^2*n!}{i!(n-i)!} = n*(n+1)*2^{n-2}$ I most probably have to use induction, but as much as I've tried, it doesn't bring me closer to a solution. I'm probably ...
3
votes
3answers
96 views

Sigma notation question

Find the value of the sum. $$\sum_{i=1}^n i(i+1)(i+2)$$ Does this mean that the answer is $$1(1+1)(1+2) + \cdots + n(n+1)(n+2)$$? Is there no value to the answer?
1
vote
1answer
27 views

Summation for computing distance with time?

I have a homework problem for Calc II that goes something like this: A tortoise and a hare are in a 1500m race. The hare goes 1m in the first second, (999/1000)m in the second second, a (999/1000)^2 ...
0
votes
2answers
32 views

How to write product of three sums

I know that by the binomial theorem, $\displaystyle \left(\sum_{n=0}^\infty a_nx^n \right)\left(\sum_{n=0}^\infty b_nx^n \right)= \sum_{n=0}^\infty \left(\sum_{k=0}^n a_kb_{n-k} x^n\right)$. How do I ...
2
votes
4answers
121 views

Prove that $1+a+a^2+\cdots+a^n=(1-a^{n+1})/(1-a)$.

I have problem. Prove this using Mathematical Induction. I am a newbie in Mathematics. Please help me. $$1+a+a^2+\cdots+a^n = \frac{1-a^{n+1}}{1-a}$$ This is my way for get the proof Basic ...
0
votes
3answers
77 views

Sum of the series $\frac{(-3)^{n-1}}{8^n}$

It might looks obvious to you but I don't manage to find the sum: $$\sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{8^n}$$ Can anyone help me ? Thanks
3
votes
1answer
166 views

Proving combinatorial identity

I need to prove following combinatorial identities: $$ \sum\limits_s(-1)^s\binom{p+s-1}{s}\binom{2m+2p+s}{2m+1-s}2^s=0 $$ $$ ...
6
votes
5answers
212 views

How can I express the sum of $\sin a+\sin2a+\sin3a+\cdots+\sin(n-1)a$?

I want to sum up the partials of a harmonic series, how do I do it? If I was using the 'Lagrange trigonometric identity to solve this problem', how would I plot it on Wolfram mathematica (using which ...
2
votes
1answer
90 views

How to calculate the integer part of the value of the following equation?

How to calculate the integer part of the value of the following equation? $$y=1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+\ldots+\frac{1}{\sqrt{1000000}}$$ It should be calculated in ...
0
votes
1answer
26 views

how do I compute this summation of the expected value

How do I compute the summation at the end: $$E(x) = \sum_{x=1}^\infty x.P(X=x) = \sum_{x=1}^{\infty} x \left(\frac{5}{6}\right)^{x-1}$$
4
votes
2answers
78 views

Prove by mathematical induction that $\sum_{i=1}^{n}\frac{i}{2^i}\leq2$ for $n\ge 1$

I have this exercise by my professor that I have no idea how to solve. Any help would be greatly appreciated: Using the method of mathematical induction show that for all $n \geq 1$, $n ...
0
votes
1answer
58 views

How to calculate $\sum_{i = 0}^{k-1}\left(\frac{5}{6}\right)^i$

I need to calculate the complexity of an algorithm. I have come across this summation that I can't evaluate, I am stuck. I haven't seen one of those in years, therefore I am rusty and really ...
4
votes
2answers
89 views

Finding the value of ${\mathop{\sum\sum\sum\sum}_{0\le i\lt j\lt k\lt l\le n }} \,n$

Finding the value of $\mathop{\sum\sum\sum\sum}_{0\le i\lt j\lt k\lt l\le n } n$ The answere is $n\cdot({^{n+1}C_4})$ My problem is to deal with the limits.
1
vote
1answer
42 views

Canceling factorials and exponentials in sum

I'm trying to understand the following the proof. I want to show that $$E\left[\frac{1}{X+1}\right] = \frac{1}{(n+1)p}(1-(1-p)^{n+1})$$ The proof goes like this: $$ \begin{align} ...
0
votes
1answer
65 views

Proof by induction that $1^2 + 2^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}$ [duplicate]

How would I go about solving this question? Use induction to prove that for all integers $n ≥ 1$, $$1^2 + 2^2 + 3^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}$$
0
votes
0answers
47 views

Write a function for the sum of a series at n given an arbitrary sequence.

I have a sequence represented by $a_n = n^2 + 3$. I want to find a function $S(n)$ which returns the first $n$ terms added together. For an arbitrary sequence given only by $a_n$ (an expression for ...
1
vote
6answers
90 views

How do I start this summation (double series?)

I'm unfamiliar with this type of problem, but I've been asked to write out the result of n = 8. The problem: Prove that $$ \sum_{i = 1}^{n} \sum_{j = 1}^{i} f(i, j) = \sum_{j = 1}^{n} \sum_{i = ...
1
vote
1answer
137 views

Double summation.

I'm in the middle of an assignment, and I'm not looking for too much help, just more of a push in the right direction (as I haven't really encountered this in my mathematics courses before). I'm ...
0
votes
1answer
60 views

Inequality with a sum and factorial

For a homework assignment we have the following question that I'm stuck on. Let $ 0 \leq y \leq 1 $ be given. $\forall m \in \mathbb{N}$, define $ \displaystyle S_m(y)=\sum_{k=0}^m \binom{m}{k}y^k$. ...
2
votes
1answer
122 views

approximation of sum of gaussian-like function?

Let: $g(u; x,s) = \dfrac{1}{s\sqrt{2\pi}} \exp\left(-\dfrac{1}{2} \left(\dfrac{x-u}{s}\right)^2\right)$ Where $x,s$ are parameters I'm looking for a closed-form solution or approximation of: ...