All Questions
1,809
questions
-1
votes
1
answer
46
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Resources to master summation symbol [closed]
I noticed that I have some difficulties to use the summation tools( change of index, double or multiples summation...). Do you have some resources or book to master this topic. I am using concrete ...
- 9
1
vote
0
answers
69
views
Evaluating an infinite series with a function
There is an infinite series, I want to transform it into a function, with digamma functions or something else. I hope someone can provide some guidance and suggestions.
$$
E(x,y)=\sum_{n=-\infty}^{\...
- 21
6
votes
2
answers
256
views
Problematic limit $\epsilon \to 0 $ for combination of hypergeometric ${_2}F_2$ functions
In an earlier question, the integral $$I_n(c)=\int_0^\infty x^n (1+x)^n e^{-n c x^2} dx$$ was considered with particular focus on its behavior for positive integer $n$. In trying to analyze this, it ...
- 16.7k
3
votes
0
answers
48
views
How to solve $\sum\limits_{n=1}^\infty \prod\limits_{k=0}^{n-1} \frac{\alpha+k}{\beta+k}$? [duplicate]
This problem: $S:=\sum\limits_{n=1}^\infty \prod\limits_{k=0}^{n-1} \frac{\alpha+k}{\beta+k}$ where $\beta > a+1, \ \ \alpha, \beta >0$ is in my problem book and I couldn't solve it
I tried to ...
- 6,563
0
votes
1
answer
60
views
Cool identities/properties involving the Alternating Harmonic Numbers
Using the following analytic continuation for the Alternating Harmonic Numbers ($\bar{H}_x=\sum_{i=1}^x\frac{(-1)^{i+1}}i$): $$\bar{H}_x=\ln2+\cos(\pi x)\left(\psi(x)-\psi\left(\frac x2\right)-\frac1x-...
- 6,549
0
votes
2
answers
89
views
A threshold for an exponential sum
I came across a sum where I have to find the smallest $n$ so that
$$\sum_{x = 0}^n \frac{250^x}{x!} \ge \frac{e^{250}}{2}$$
I wrote a Java code and the result was 55 but with Desmos it was over 129 (...
1
vote
0
answers
41
views
A partial sum formula [duplicate]
I'm very familiar with partial sums and such little bit hard once, but I was wondering is there a partial sum formula for that
$$\displaystyle\sum_{n=1}^k n^n$$
I have tried with Wolfram alpha but I ...
- 21
1
vote
0
answers
81
views
Closed form for $ \sum_{a_1=0}^\infty~\sum_{a_2=0}^\infty~\cdots~\sum_{a_n=0}^\infty \dfrac1{(a_1!+a_2!+\ldots+a_n!)} $ [closed]
After reading this post and the general solution for that case, I wonder if there is a closed form for the general solution for this sum:
$ \sum_{a_1=0}^\infty~\sum_{a_2=0}^\infty~\cdots~\sum_{a_n=0}^\...
- 923
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
2
votes
2
answers
80
views
Evaluate $\sum_{k=1}^{\infty}\frac{9k-4}{3k(3k-1)(3k-2)}$
We want to evaluate the series: $$\sum_{k=1}^{\infty}\frac{9k-4}{3k(3k-1)(3k-2)}$$
My try :
We have :
$$\frac{9k-4}{3k(3k-1)(3k-2)}=\frac{1}{3k-1}+\frac{1}{3k-2}-\frac{2}{3k}$$
Therefore:
$$\sum_{k=1}^...
- 2,348
1
vote
2
answers
57
views
Summation form of improper integrals
On page 9, Edwards has this expression
$$ \int_0^{\infty} e^{-nx} x^{s-1} dx = \frac{\Pi(s-1)}{n^s}$$
obtained from Euler’s factorial formula by replacing $x$ with $nx$. Can you help with the next ...
- 447
2
votes
2
answers
78
views
How to calculate thi sum $\sum_{n=2}^{\infty} \frac{\left( \zeta(n) - 1 \right) \cos \left( \frac{n \pi}{3} \right)}{n}$
My question
$$ \displaystyle{\mathcal{S} = \sum_{n=2}^{\infty} \frac{\left( \zeta(n) - 1 \right) \cos \left( \frac{n \pi}{3} \right)}{n}}$$
My try to solve the integral
$$\displaystyle \sum\limits_{n =...
- 121
0
votes
0
answers
14
views
Getting the formular of a summation [duplicate]
im kind of stuck at my math homework from my calculus class. To progress further i need to be able to write a Summation into a forumular(?), as seen in the picture. Is there any proven method to do ...
- 1
0
votes
1
answer
95
views
Showing $\sum_{n=1}^{\infty }\left ( \sum_{j=1}^{\infty }\frac{x^{(n-j)^2}-x^{(n+j-1)^2}}{(2n-1)(2j-1)} \right ) = \frac{\pi^2}{8}$
Show that
$$\sum_{n=1}^{\infty }\left ( \sum_{j=1}^{\infty }\frac{x^{(n-j)^2}-x^{(n+j-1)^2}}{(2n-1)(2j-1)} \right) = \frac{\pi^2}{8}$$
I liked this problem because the result is a final answer, and ...
- 1,429
1
vote
5
answers
113
views
Alternative ways to evaluate $\sum_{k=1}^{n}(2k+1)^2$
I'm looking for alternative ways to calculate $$\sum_{k=1}^{n}(2k+1)^2$$
The normal approach is to expand $(2k+1)^2$ and use the formulas $\sum_{k=1}^n k^2 = \dfrac{n(n+1)(2n+1)}6$ , $\sum_{k=1}^n k = ...
- 6,794
0
votes
4
answers
196
views
How to evaluate $\sum\limits_{n=3}^ \infty \frac{1}{n \ln(n)(\ln(\ln(n)))^2}$
I saw this problem : Prove that $\sum\limits_{n=3}^ \infty \frac{1}{n \ln(n)(\ln(\ln(n)))^2}$ converges, this is an easy problem could be proved using Cauchy condensation test twice.
$$\sum_{n=3}^ \...
- 6,563
0
votes
1
answer
47
views
Sum sequence using Stolz–Cesàro
I have this sequence, and I need to find the convergence of the sum sequence.
The answer is - sum equal π/4.
But I tried to solve it by Stolz–Cesàro, as you can see in the picture, And what I got is ...
- 215
1
vote
1
answer
46
views
Simplifying $\frac{1}{2}\sum_{n=0}^{\infty}{(n+1)(n+2)(z^n+z^{n+1})}$
This is part of a larger problem where I am trying to prove $\sum_{n=0}^{\infty}{(n+1)^2z^n}=(z+1)/(1-z)^3$. Thus far I have used the derivatives of the geometric series to obtain $\frac{1}{2}\sum_{n=...
- 142
4
votes
1
answer
125
views
Find value of this sum
Let $$\lim_{x\rightarrow 0}\frac{f^{}(x)}{x}=1$$
and for every $x,y \in \mathbb{R} $ we have:
$$f(x+y)=f(x)-f(y)+ xy(x+y)$$
Now Find :
$$\sum_{i=11}^{17}f^{\prime} (i)$$
I think this question is ...
- 4,687
0
votes
2
answers
200
views
how to calculate $\sum\limits_{k=1}^{+\infty }{\arctan \frac{1}{1+k^{2}}}$
Question: how to calculate $$\sum\limits_{k=1}^{+\infty }{\arctan \frac{1}{1+k^{2}}}$$
My attempt
Let
$\arctan \theta =\frac{i}{2}\ln \left( \frac{i+\theta }{i-\theta } \right)$
$$S=\sum\limits_{k=1}^{...
2
votes
1
answer
177
views
Show that $ \int_0^{\pi\over 2}\frac{\sin(2nx)}{\sin^{2n+2}(x)}\frac{1}{e^{2\pi \cot x}-1}dx =(-1)^{n-1}\frac{2n-1}{4(2n+1)} $
Show that $$ \int_0^{\pi\over 2}\frac{\sin(2nx)}{\sin^{2n+2}(x)}\frac{1}{e^{2\pi \cot x}-1}dx =(-1)^{n-1}\frac{2n-1}{4(2n+1)}
$$
My attempt
Lemma-1
\begin{align*}
\frac{\sin(2nx)}{\sin^{2n}(x)}&=\...
0
votes
0
answers
132
views
Calculation of $\sum_{n=1}^{\infty}\frac{\psi_1(n)}{2^nn^2}$
Calculation of $$\sum_{n=1}^{\infty}\frac{\psi_1(n)}{2^nn^2}$$
My attempt
\begin{align*}
\sum_{n=1}^\infty\frac{\psi_1(n)}{2^n n^2} &= -\sum_{n=1}^\infty\psi_1(n)\left(\frac{\log(2)}{2^n n}+\int_0^...
4
votes
2
answers
203
views
How to evaluate this sum $\sum_{n=1}^{\infty} \frac{(-1)^n}{(n^2 + 3n + 1)(n^2 - 3n + 1)}$
How to evaluate this sum $$\sum_{n=1}^{\infty} \frac{(-1)^n}{(n^2 + 3n + 1)(n^2 - 3n + 1)}$$
My attempt
$$\sum_{n=1}^{\infty} \frac{(-1)^n}{(n^2 + 3n + 1)(n^2 - 3n + 1)}$$
$$= \sum_{n=1}^{\infty} \...
8
votes
2
answers
244
views
How to calculate $\int _0^1 \int _0^1\left(\frac{1}{1-xy} \ln (1-x)\ln (1-y)\right) \,dxdy$
Let us calculate the sum
$$
\displaystyle{\sum_{n=1}^{+\infty}\left(\frac{H_{n}}{n}\right)^2},
$$
where $\displaystyle{H_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n}}$ the $n$-th harmonic number.
My try
The ...
1
vote
1
answer
123
views
Convergence of summation of complex exponentials with alternating exponent
Related to my previous question, consider $$f(s)=\sum_{k=1}^{\infty} \exp(-s(-2)^k)$$where $s\in\mathbb{C}$ is a complex number. According to the Willie Wong's comment, $f(s)$ diverges when $\Re\{s\} \...
- 4,359
2
votes
2
answers
228
views
Evaluate the infinite product $ \prod_{n=1}^{\infty} \left ( 1 + \frac{x^2}{n^2+n-1} \right )$
Question statement
Evaluate the infinite product
$$\displaystyle{\prod_{n=1}^{\infty} \left ( 1 + \frac{x^2}{n^2+n-1} \right ) }$$
My try
Because of the square of $\displaystyle{x}$ , we can consider $...
1
vote
1
answer
130
views
Borel Regularization of $\sum_{n=1}^\infty \ln(n)$ [closed]
I'm trying to solve the following taylor series
$$\sum_{n=0}^\infty \frac{x^n}{n!} \ln(n+1)$$
so I can regularize the following sum
$$\sum_{n=1}^\infty \ln(n)$$
Using Borel Regularizaiton I can use ...
0
votes
1
answer
41
views
How to express $\sum_{i=0}^{m} \exp [(\frac{a}{b+c+i})^2] $ in terms of an integral?
I have this sum
$$\sum_{i=0}^{m} \exp [(\frac{a}{b+c+i})^2] $$
where the upper limit $m$ is a finite non-negative integer, and $a,b,c\in\mathbb{R}$. I want to transform summation to an integral ...
- 51
0
votes
0
answers
33
views
Question on transforming a sum to an integral using the Euler–Maclaurin formula.
I have a question regarding transforming a summation to an integral using the Euler–Maclaurin formula. Imagine I have this sum
$$\sum_{i=0}^{m} f(i) \qquad \text{with} \qquad f(i)= \exp [(\frac{a}{b+...
- 51
1
vote
0
answers
69
views
Leibniz integral rule for summation
Context
Fundamental points of Feymann trick:
You have an integral $I_0=\int_a^b f(t)\mathrm{d}t$
Now consider a general integral $I(\alpha)=\int_a^b g(\alpha,t)\mathrm{d}t$ so that $I'(\alpha)=I_0$ ...
2
votes
0
answers
67
views
Closed form for $\psi^{1/k}(1)$, where $k$ is an integer
I have proven the identity
$$
\sum_{k=1}^{\infty} \dfrac{\operatorname{_2F_1}(1, 2, 2-1/t,-1/k)}{{k}^{2}} = Γ(2-\dfrac{1}t){\psi^{1/t}(1)}+\psi(-\dfrac{1}t)(\dfrac{1}t(1-\dfrac{1}t))+\gamma(1-\dfrac{1}...
0
votes
0
answers
104
views
Is this divergent series, convergent?
Examining the series $\sum_{n=1}^{\infty} \frac{1}{nx}$ alongside its integral counterpart reveals insights into its convergence. Notably, the integral over intervals from $10^n$ to $10^{n+1}$ yields ...
- 81
1
vote
2
answers
82
views
Upper rectangle area sum to approximate 1/x between $1\leq x\leq 3$
I am trying to figure out how to use rectangles to approximate the area under the curve $1/x$ on the interval $[1,3]$ using $n$ rectangle that covers the region under the curve as such.
Here is what I ...
- 1,071
2
votes
2
answers
99
views
Prove $\frac12\left(\psi\left(\frac{x+1}2\right)-\psi\left(\frac x2\right)\right)=\psi(x)-\psi\left(\frac x2\right)-\ln2$
Desmos suggests that$$\frac12\left(\psi\left(\frac{x+1}2\right)-\psi\left(\frac x2\right)\right)=\psi(x)-\psi\left(\frac x2\right)-\ln2$$Where $\psi$ is the digamma function. I can write the LHS as $$\...
- 6,549
0
votes
0
answers
31
views
Can we compare the arithmetic mean of the ratios given the comparison between individual arithmetic means?
I have positive real random numbers $u_1,\ldots,u_n$ and $v_1,\ldots,v_n$ and $x_1,\ldots,x_n$ and $y_1,\ldots,y_n$.
I know that the arithmetic mean of $u_i$'s is greater than the arithmetic mean of $...
- 452
1
vote
1
answer
95
views
Evaluation of $\int_{0}^{\frac{\pi}{4}} \frac{\log(\log(\tan(\frac{\pi}{4} + x))) \cdot \log(\tan(\frac{\pi}{4} + x)))}{\tan(2x)} \,dx$ [closed]
$$\int_{0}^{\frac{\pi}{4}} \frac{\log(\log(\tan(\frac{\pi}{4} + x))) \cdot \log(\tan(\frac{\pi}{4} + x)))}{\tan(2x)} \,dx$$
$$\int_{0}^{\frac{\pi}{4}} \frac{\log(\log(\tan(\frac{\pi}{4} + x))) \cdot \...
2
votes
0
answers
365
views
A sum of two curious alternating binoharmonic series
Happy New Year 2024 Romania!
Here is a question proposed by Cornel Ioan Valean,
$$\sum_{n=1}^{\infty}(-1)^{n-1} \frac{1}{2^{2n}}\binom{2n}{n}\sum_{k=1}^n (-1)^{k-1}\frac{H_k}{k}-\sum_{n=1}^{\infty}(-1)...
- 5,495
5
votes
2
answers
157
views
Show that $\sum_{n=1}^{\infty} \frac{\binom{2n}{n} (H_{2n} - H_n)}{4^n (2n - 1)^2} = 2 + \frac{3\pi}{2} \log(2) - 2G - \pi$
Show that $$\sum_{n=1}^{\infty} \frac{\binom{2n}{n} (H_{2n} - H_n)}{4^n (2n - 1)^2} = 2 + \frac{3\pi}{2} \log(2) - 2G - \pi$$
My try :
We know that
$$\sum_{n=1}^{\infty} \binom{2n}{n} (H_{2n} - H_{n}) ...
1
vote
1
answer
131
views
for any positive numbers $p_k$ how to find the minimum of $\sum_{k=1}^n a_k^2 +(\sum_{k=1}^n a_k)^2$ when $\sum_{k=1}^n p_ka_k=1$? [duplicate]
For any positive numbers $p_k$ how to find the minimum of $\sum\limits_{k=1}^n a_k^2 +(\sum\limits_{k=1}^n a_k)^2$ when $\sum\limits_{k=1}^n p_ka_k=1$?
I saw this problem on my problem book and I ...
- 6,563
1
vote
1
answer
129
views
Prove $\sum_{k=j}^{\lfloor n/2\rfloor}\frac1{4^k}\binom{n}{2k}\binom{k}{j}\binom{2k}{k}=\frac1{2^n}\binom{2n-2j}{n-j}\binom{n-j}j$
let $x>1$, $n\in \mathbb N$ and
$$P_{n}(x)=\dfrac{1}{\pi}\int_{0}^{\pi}(x+\sqrt{x^2-1}\cos{t})^ndt.$$
Prove that
$$P_{n}(x)=\dfrac{1}{\pi}\int_{0}^{\pi}\dfrac{1}{\left(x-\sqrt{x^2-1}\cos{t}\right)^{...
- 137
0
votes
2
answers
80
views
On the convergence of $\sum_{n=0}^\infty(f(x)-T_n\{f\}(x))$
To test the efficiency of the Taylor Series at approximating functions, I was wondering whether$$\sum_{n=0}^\infty(f(x)-T_n\{f\}(x))\tag{$\star$}$$converges, where $T_n\{f\}(x)$ is the degree $n$ ...
- 6,549
0
votes
1
answer
142
views
How to rigorously prove that $e^x = \sum\limits_{n=0}^ \infty \frac{x^n}{n!}$ without defining derivatives? [duplicate]
In my problem book, there was a question: By defining $e= \lim\limits_{n \to \infty}\left( 1+\frac{1}{n} \right) ^n$ prove that $e^x = \sum\limits_{n=0}^ \infty \frac{x^n}{n!}$. this is a strange ...
- 6,563
22
votes
3
answers
1k
views
Conjecture: $\sum\limits_{k=1}^nk^m=S_3(n)\times\frac{P_{m-3}(n)}{N_m}$ for odd $m>1 \ ;\ =S_2(n)\times\frac{P_{m-2}'(n)}{N_m}$ for even $m$.
When I was in high school, I was fascinated by $\displaystyle\sum\limits_{k=1}^n k= \frac{n(n+1)}{2}$ so I tried to find the general solution for $\displaystyle\sum\limits_{k=1}^n k^m$ s.t $m \in \...
- 6,563
-1
votes
1
answer
81
views
$\sum _{k=1}^{\infty }{\frac {\coth(k\pi )}{(k\pi )^{4n-1}}}$ [duplicate]
Show that
$${\displaystyle \sum _{k=1}^{\infty }{\frac {\coth(k\pi )}{(k\pi )^{4n-1}}}=\sum _{k=0}^{2n}(-1)^{k-1}\,{\frac {\zeta (2k)}{\pi ^{2k}}}\,{\frac {\zeta (4n-2k)}{\pi ^{4n-2k}}}\qquad n\in \...
user1235604
1
vote
2
answers
205
views
Which closed form expression for this series involving Catalan numbers : $\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{4^nn^2}\binom{2n}{n}$
Obtain a closed-form for the series: $$\mathcal{S}=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{4^nn^2}\binom{2n}{n}$$
From here https://en.wikipedia.org/wiki/List_of_m ... cal_series we know that for $\...
6
votes
4
answers
241
views
Evaluate $\int_{0}^{1}\{1/x\}^2\,dx$
Evaluate
$$\displaystyle{\int_{0}^{1}\{1/x\}^2\,dx}$$
Where {•} is fractional part
My work
$$\displaystyle{\int\limits_0^1 {{{\left\{ {\frac{1}{x}} \right\}}^2}dx} = \sum\limits_{n = 1}^\infty {\...
user1235604
2
votes
1
answer
78
views
$\frac{(1+x)^n}{(1-x)^3}=a_{0}+a_{1}x+a_{2}x^2+\cdots$ show that ${a_{0}+\cdots+a_{n-1}=\frac{n(n+2)(n+7)2^{n-4}}{3}}$
$$\displaystyle{\frac{(1+x)^n}{(1-x)^3}=a_{0}+a_{1}x+a_{2}x^2+\cdots}$$, show that $$\displaystyle{a_{0}+\cdots+a_{n-1}=\frac{n(n+2)(n+7)2^{n-4}}{3}}$$
When i gave this problem to my friends they said ...
user1235604
2
votes
1
answer
212
views
Calculate the value $\lim_{n\to \infty}\frac{\sum_{j=1}^n \sum_{k=1}^n k^{1/k^j}}{\sqrt[n]{(\sum_{j=1}^n j!)\sum_{j=1}^n j^n}}$
As in title, I want to calculate the following value $$\lim_{n\to \infty}\frac{\sum_{j=1}^n \sum_{k=1}^n k^{1/k^j}}{\sqrt[n]{(\sum_{j=1}^n j!)\sum_{j=1}^n j^n}}.$$
Here is my attempt:
Since $\sum_{j=...
- 312
-1
votes
2
answers
96
views
Evaluate $\sum_{m=1}^{\infty} \frac{(-1)^m \mathcal{H}_{2m} }{2m+1} - \frac{1}{2} \sum_ {m=1}^{\infty} \frac{(-1)^m \mathcal{H}_m}{2m+1}$ [duplicate]
Let's declare $\mathcal{G}$ is constant of Catalanand the $\mathcal{H}_m-st$ mharmonic term. Let it be shown that:
$$\displaystyle{\sum_{m=1}^{\infty} \frac{(-1)^m \mathcal{H}_{2m} }{2m+1} -\frac{1}{2}...
1
vote
2
answers
105
views
Evaluate $\sum_{n=1}^{\infty} (-1)^{n+1} H_n \left( \frac{1}{n+1} - \frac{1}{n+3} + \frac{1}{n+5} - \ldots \right)$
$$\sum_{n=1}^{\infty} (-1)^{n+1} H_n \left( \frac{1}{n+1} - \frac{1}{n+3} + \frac{1}{n+5} - \ldots \right) = \frac{\pi}{16} \cdot \log(2) + \frac{3}{16} \cdot \log(2) - \frac{\pi^2}{192}$$
$$\sum_{k=...