All Questions
47
questions
0
votes
1
answer
77
views
Please help me to find the sum of an infinite series. [duplicate]
Please help me to solve this problem. I need to find the sum of an infinite series:
$$
S = 1 + 1 + \frac34 + \frac12 + \frac5{16} + \cdots
$$
I tried to imagine this series as a derivative of a ...
- 21
0
votes
1
answer
117
views
Minimising problem with sums and matrices
We assume that $p_0,...,p_k$ are pairwise conjugate to Q and we define a function $f(x)=\frac{1}{2}x^TQx+g^Tx$, and then we have to show that the solution $\alpha_0,...,\alpha_k$ is $\alpha_k=-\frac{...
- 558
1
vote
1
answer
94
views
Finite sum of csc via sum of cot [closed]
Would anyone give a clue how to prove the following identity:
$\sum_{k =1}^{n-1} \csc\left(\frac{k\pi}{n}\right) = -\frac{1}{n}\sum_{k=0}^{n-1}(2k+1)\cot\left(\frac{(2k+1)\pi}{2n}\right)$.
I tried ...
- 19
0
votes
1
answer
145
views
Calculating an infinite sum and finding a limit
Is it possible to evaluate or find the solution to the following infinite sum with an inverse trigonometric function (arctan), and its limit?
$\displaystyle\lim_{x\to\infty}\frac{1}{\sqrt x}\sum_{n=1}^...
- 293
1
vote
1
answer
99
views
Upper bound on the integral of a step function inequality
Problem
I have been working through content in real analysis and came across across an inequality in a proof that is unclear to me.
We are told that a function $\psi$ is a step function on the ...
- 4,331
3
votes
2
answers
221
views
Convergence of $\sum_{n=0}^{\infty} \frac{4^n}{3^n+7^n}$
I need help with this.
$\sum_{n=0}^{\infty} \frac{4^n}{3^n+7^n}$
I know that it converges but i can not proove why.
I tried to rewrite it, it seems to be a geometric serie. I tried to do a common ...
- 47
1
vote
0
answers
343
views
How to find tight upper bound?
I am trying to find a tight upper bound on $\sum_{i=1}^nf(s_i)\left(\frac{1 - \exp(-s_it)}{s_it} - \frac{1}{1 + s_it}\right)$ where the maximum value of function is $a$ and minimum is $b$, and both of ...
- 81
0
votes
1
answer
462
views
The Heaviside functions in the following?
For $(\mu_{i})_{i=\overline{1,n}}$ are real positive parameters, we have $H$ is the Heaviside function, i.e
$$\forall i = \overline{1,n}, ~~~~H(u-\mu_i)=\left\{\begin{array}{ll} 1 & \quad \mbox{if ...
- 37
1
vote
1
answer
116
views
How do I obtain the Jacobi theta function?
If I enter the series $$\sum_{n=-\infty}^\infty e^{-(n+x)^2}$$ into WolframAlpha it expresses it by the Jacobi theta function $$\sqrt{\pi} \vartheta_3(\pi x,e^{-\pi^2})=\sqrt{\pi} \big(1+2\sum_{n=1}^\...
- 139
0
votes
1
answer
66
views
Closed form of this series
I am wondering if the following series has a closed form expression:
$\sum_{x=0}^{\infty} \frac{g(x)\lambda^x}{x!}$, where $g(x)=\frac{x}{2}$ for even $x$, and $g(x)=\frac{x-1}{2}$ for odd $x$.
I ...
- 284
1
vote
1
answer
53
views
Sum of the absolute values
I was going over a question and need your opinion about solution of the sum of the absolute values as
$$S_n = |0-a|+|1-a|+|2-a|+ \dots + |(n-1)-a|+|n-a|$$
where a is a constant term. What could be ...
- 21
1
vote
2
answers
56
views
Problems with understanding a given Sum Identity
In a textbook I found the following equation without any explaination:
\begin{align}
\sum_{\substack{j, k=1,\\ j\neq k}}^n\frac{1}{(x-x_j)(x-x_k)} - \frac{3}{2}\left( \sum_{j=1}^n \frac{1}{x-x_j}\...
- 974
2
votes
3
answers
53
views
Simple question about a formula for sums
Why is
$$
\left(\sum_{i=1}^{n}a_i\right)^2= \sum_{i=1}^{n}\sum_{j=1}^{n}a_ia_j
$$
I guess it is fairly clear when you do it for $n=2$ for example, but I can't really proof it for all $n$ with ...
- 1,652
0
votes
1
answer
71
views
Finding the infinite sum using Leibniz Test
I have been given a task in my previous lecture to determine whether the infinite sum;
$$\sum_{n=1}^\infty \frac{\cos (\pi n)\ln n}{n}$$
Converges or diverges.
My perspective on the problem is that ...
- 1,136
1
vote
1
answer
49
views
Continous function $r(x)$ on $[0,1]$ such that $\lim_{n\rightarrow\infty}\sum_{k=0}^{n}\frac{1}{n}g(\frac{k}{n},a,b)r(\frac kn)=0$
The problem is related to this question (https://math.stackexchange.com/posts/3073378/edit).
Is there any general way to construct a non-zero continous function $r(x)$ on $[0,1]$, independent of ...
- 1,259
2
votes
0
answers
55
views
Closed form for a series involving the $\Gamma$ and $\zeta$ functions
I was just wondering wether one can derive a closed form for $$\sum_{n=1}^{\infty}\frac{1}{\Gamma\left(\frac{1}{n}\right)\zeta\left(1+\frac{1}{n}\right)}$$
Numerical simulation gives $S=1.20154...$
...
- 1,984
0
votes
1
answer
30
views
How much a variable contributed to the result of some cost function
I have this simple cost function: $\sum_{i=1}^n d_i\times h_i \times a_i$
I wanted to analyze, for example, how much the $a$ component/variable contributed to the final cost function. In other ...
3
votes
4
answers
192
views
How to prove that $1+\frac11(1+\frac12(1+\frac13(...(1+\frac1{n-1}(1+\frac1n))...)))=1+\frac1{1!}+\frac1{2!}+\frac1{3!}+...+\frac1{n!}$?
I'm trying to prove that
$$1+\frac11(1+\frac12(1+\frac13(...(1+\frac1{n-1}(1+\frac1n))...)))=1+\frac1{1!}+\frac1{2!}+\frac1{3!}+...+\frac1{n!}$$
Using induction, suppose that
$$1+\frac11(1+\frac12(...
user547800
1
vote
0
answers
86
views
Is the following inequality True or False? [duplicate]
Ι got a feeling that $$\sum_{x=1}^{\infty}\Big\lvert\sum_{k=0}^{\infty} \frac{x^{2k}}{(k+1)!}(-1)^{k} \Big\rvert \geq \sum_{n=1}^{\infty} \frac{1}{n} $$
because $$\sum_{x=1}^{\infty} \Big\lvert x-\...
- 2,782
1
vote
1
answer
35
views
find a where $\sum_{n=n(a)}^\infty (-1)^n \frac{(2+n^a)}{n}$ converges and absolutely converges
Study $a$ so that the series convergences and absolute convergences
I just know that $a$ is positive
I applied the ratio test, I've found that $0<a<1$
I have no idea for absolute convergence.
...
3
votes
2
answers
82
views
How to find the limit:$\lim_{n\to \infty}\left(\sum_{k=n+1}^{2n}\left(2(2k)^{\frac{1}{2k}}-k^{\frac{1}{k}}\right)-n\right)$
How to find the limit:$$\lim_{n\to \infty}\left(\sum_{k=n+1}^{2n}\left(2(2k)^{\frac{1}{2k}}-k^{\frac{1}{k}}\right)-n\right)$$
I can't think of any way of this problem
Can someone to evaluate this?
...
- 1,431
2
votes
1
answer
232
views
Summation notation two sums?
$$\sum_{n=1}^{2^k - 1}\frac{1}{n} = \sum_{n=1}^{2-1}\frac{1}{n} + \sum_{n=2}^{2^2 - 1}\frac{1}{n} + \cdots + \sum_{n = 2^{k-1}}^{2^k - 1}\frac{1}{n} = \boxed{\sum_{j = 0}^{k-1}\sum_{n = 2^j}^{2^{j+1} -...
- 513
0
votes
1
answer
72
views
A Finite Summation
Is there an easy way to find the following summation (question created by Priyanshu Mishra on Brilliant.org, but now has been deleted):
$$\sum _{n=2}^{25}
\frac{(n+1)!-n!}{n^{18}+n^{17}+n^{15}}$$
...
user485639
4
votes
2
answers
212
views
Asymptotic of a sum: $\sum_{k=n}^{\infty}{f(k)} = \int_{n}^{\infty}{f(t)dt} +\frac{f(n)}{2}+\mathcal{O}(f'(n))$
Someone told me that the following formula holds for $f$ differentiable and decreasing, with $\lim_{x\rightarrow +\infty}{f(x)}=0$.
$$\sum_{k=n}^{\infty}{f(k)} = \int_{n}^{\infty}{f(t)dt} +\frac{f(...
- 301
1
vote
2
answers
487
views
Finding the limit as $k$ tends to infinity of this sum
$\sum_{i=0}^{\lceil zk \rceil}{k\choose i} p^i(1-p)^{k-i}$
$z, p \in [0,1]$
I am looking to find the limit as $k$ tends to infinity but don't know how I would do this
- 125
3
votes
1
answer
612
views
Changing the order of summation and using a change of variables
I'm looking at the below, and I don't understand how to get the second equality using the change of variables $m=k-l$. I've tried changing the order of summation to get $$\sum_{k=2^j+1}^{2^{j+1}}\sum_{...
- 5,611
3
votes
2
answers
2k
views
Partial sums of $nx^n$
WolframAlpha claims:
$$\sum_{n=0}^m n x^n = \frac{(m x - m - 1) x^{m + 1} + x}{(1 - x)^2} \tag{1}$$
I know that one can differentiate the geometric series to compute $(1)$ when it is a series, i.e. $m=...
- 11.8k
4
votes
2
answers
460
views
What's a closed form for $\sum_{k=0}^n\frac{1}{k+1}\sum_{r=0\\r~is~odd}^k(-1)^r{k\choose r}r^n$
I want to use a closed form of
$$\sum_{k=0}^n\frac{1}{k+1}\sum_{\ \ \ r=0\\r\text{ is odd}}^k(-1)^r{k\choose r}r^n$$
and
$$\sum_{k=0}^n\frac{1}{k+1}\sum_{\ \ \ r=0\\r\text{ is even}}^k(-1)^r{k\choose ...
- 30.1k
7
votes
2
answers
317
views
Stuck with this limit of a sum: $\lim _{n \to \infty} \left(\frac{a^{n}-b^{n}}{a^{n}+b^{n}}\right)$.
Here's the limit: $$\lim _{n \to \infty} \left(\frac{a^{n}-b^{n}}{a^{n}+b^{n}}\right)$$
The conditions are $b>0$ and $a>0$.
I tried this with the case that $a>b$:
$$\lim _{n \to \infty} \...
- 361
0
votes
2
answers
107
views
Calculate (i.e. express without using infinite sum): $\frac{1}{1!\cdot1}+\frac{2}{2!\cdot2}+\frac{4}{4!\cdot4}+\frac{8}{6!\cdot8}+..$
Calculate (i.e. express without using infinite sum):
$$\frac{1}{1!\cdot1}+\frac{2}{2!\cdot2}+\frac{4}{4!\cdot4}+\frac{8}{6!\cdot8}+..$$
In sum it would be:
$$\sum_{n=0}^{\infty}\frac{2^{n}}{(...
- 1,007
1
vote
1
answer
188
views
Calculate (express without infinite sum): $\frac{2}{1\cdot3}-\frac{4}{2 \cdot 9} + \frac{8}{3\cdot 27}- \frac{16}{4 \cdot 81} + ...$
This is no homework, it's a task from an old exam and I'm wondering how it's solved correctly.
Calculate (express without infinite sum):
$$\frac{2}{1\cdot3}-\frac{4}{2 \cdot 9} + \frac{8}{3\cdot ...
- 4,710
3
votes
1
answer
175
views
For which values of $p$ the series $\sum_{n = 2}^{\infty}\frac{1}{\ln^p{n}}$ converges?
I'm trying to find all values of $p$ for which the following series converges:
$$\sum_{n = 2}^{\infty}\frac{1}{\ln^p{n}}$$
So my first approach was to use the integral test because $\frac{1}{\ln^p{n}...
- 31
2
votes
1
answer
122
views
Calculate $\sum_{k=0}^\frac{n}{2}(-1)^k{n \choose 2k}$ and $\sum_{k=0}^\frac{n-1}{2}(-1)^k{n \choose 2k+1}$
a) $\sum_{k=0}^\frac{n}{2}(-1)^k{n \choose 2k}$
b) $\sum_{k=0}^\frac{n-1}{2}(-1)^k{n \choose 2k+1}$
I think a way to calculate the sums is to see what happens to $(1+i)^n$
but after trying for 2 ...
- 65
2
votes
1
answer
368
views
How is $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}= \log 2?$ [duplicate]
How is $$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}= \log 2?$$
I haven't done sequences in a long time, therefore proving this seems almost impossible. How is this sum gotten. Help very much appreciated....
- 2,866
5
votes
3
answers
171
views
How to evaluate $\sum_{n=2}^\infty\frac{(-1)^n}{n^2-n}$
How would you go about evaluating:$$\sum_{n=2}^\infty\frac{(-1)^n}{n^2-n}$$
I split it up to $$\sum_{n=2}^\infty\left[(-1)^n\left(\frac{1}{n-1}-\frac{1}{n}\right)\right]$$
but I'm not sure what to ...
- 149
0
votes
1
answer
72
views
Using Partial Summation to evaluate a series
$$S = \sum_{x=1}^{\infty} \frac{\sin(x)}{x}$$
Using partial summation. Obviously,
$$S = \lim_{n \to \infty} \sum_{x=1}^{n} \frac{\sin(x)}{x}$$
Partial Summation:
\begin{align*}
\sum_{n=1}^{N} a(n)...
- 1,736
0
votes
1
answer
45
views
How to prove $ \sum_{k=1}^{p+1} \binom{p+1}{k}S_{n}^{p+1-k} = (n+1)^{p+1}-1$?
This is the solution $ \sum_{k=1}^{p+1} \binom{p+1}{k}S_{n}^{p+1-k} = \sum_{k=1}^{p+1} \binom{p+1}{k} \sum_{l=1}^{n}l^{p+1-k} = \sum_{l=1}^{n}(l+1)^{p+1}-l^{p+1} = (n+1)^{p+1}-1$ ?
With $S_{n}^{p} = ...
3
votes
0
answers
94
views
Can this summation be expressed differently?
Lets say I have a sum that states the following
$$
\sum_{j=0}^{k-c} {k-c \choose j}\ln(a)^{k-c-j} \frac{d^j}{dx^j}[(x)_c]
$$
where $(x)_c$ is the falling factorial such that
$$
(x)_c = x(x-1)(x-2)\...
- 1,957
4
votes
1
answer
124
views
How find this sum $\sum_{k=1}^{\infty}\frac{1}{1+a_{k}}$
Let $\{a_{n}\}$ be the sequence of real numbers defined by $a_{1}=3$ and for all $n\ge 1$,
$$a_{n+1}=\dfrac{1}{2}(a^2_{n}+1)$$ Evaluate
$$\sum_{k=1}^{\infty}\dfrac{1}{1+a_{k}}$$
My idea 1:
...
- 93.6k
0
votes
1
answer
2k
views
By applying term-wise differentiation and integration find the sum of the series $\sum_{k=1}^{\infty}\frac{x^k}{k}$
I need to find the sum of the following series: $$\sum_{k=1}^{\infty}\frac{x^k}{k}$$ on the interval $x\in[a,b], -1<a<0<b<1$ using term-wise differentiation and integration.
Can anyone ...
user184036
-1
votes
2
answers
2k
views
Integral of $\sin(x)$ using power series.
$\displaystyle \int_{0}^{1} \sin(x) \, dx$
$\sin(x) = \displaystyle \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$
Integrating this from $0 \to 1$
On the RHS we get
$\displaystyle (-)\sum_{n=...
- 11.2k
0
votes
3
answers
1k
views
Evaluate infinite sum for $\frac{1}{n^4}$ using integration
$\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^4}$
I want to evaluate this sum by the use of integration.
$\displaystyle \int_{\frac{1}{n^4}}^{\frac{4}{n^4}} 1 \space dx = \frac{4}{n^4} - \frac{1}{...
- 11.2k
3
votes
2
answers
33k
views
Finding the infinite sum of $e^{-n}$ using integrals
I am trying to understand this:
$\displaystyle \sum_{n=1}^{\infty} e^{-n}$ using integrals, what I have though:
$= \displaystyle \lim_{m\to\infty} \sum_{n=1}^{m} e^{-n}$
$= \displaystyle \lim_{m\...
- 11.2k
5
votes
1
answer
95
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 the ...
- 55
1
vote
5
answers
436
views
Finit Sum $\sum\limits_{i=1}^{100}i^8-2 i^2$
Can anyone help me?
How can I find
$$\sum_{i=1}^{100}i^8-2i^2 $$
- 331
8
votes
2
answers
170
views
Let $f(\frac ab)=ab$, where $\frac ab$ is irreducible, in $\mathbb Q^+$. What is $\sum_{x\in\mathbb Q^+}\frac 1{f(x)^2}$?
Let $f(\frac ab)=ab$, where $\frac ab$ is irreducible, in $\mathbb Q^+$. What is $\sum_{x\in\mathbb Q^+}\frac 1{f(x)^2}$?
Club challenge problem. I don't think it's possible to do with only high ...
- 143
1
vote
3
answers
333
views
Boundedness of the sum of $\arctan n$
Could someone please explain me why $$s_n=\sum^n_1 \arctan(k)$$ is bounded?
P.S I need it to understand something. Thanks beforehand.
- 11