7
votes
4answers
95 views

A closed form of $\sum_{k=0}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$

I am looking for a closed form of the following series \begin{equation} \mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right) \end{equation} I have no idea how to ...
1
vote
4answers
150 views

Find whether the following series converges or diverges $\sum_{n=1}^{\infty}\frac{\ln n }{\sqrt{n}}$

Looking for a witty answer. I can see that the given series converges by AST. All Ideas Appreciated
1
vote
2answers
95 views

How to find $\sum_{r=1}^{n} r^2\cos {(r\theta)}$

How do you find the sum $$S(\theta)=\displaystyle\sum_{r=1}^{n} r^2\cos {(r\theta)}$$? I observed that if $f(\theta)$ is $\sum\cos {(r\theta)}$, then $S(\theta)=-f''(\theta)$. Help will be ...
0
votes
2answers
22 views

How to get a partial sum formula

Let S denote sum from 1 to n of (k-1)/k! . I tried obtaining a partial sum formula, but I couldn't get too far. WolframAlpha comes with quite a simple form, but I fail to see how they got there . Can ...
3
votes
1answer
44 views

investigate $\sum\limits_{n\ge1}{\frac{(-1)^n}{n^\alpha \ln n}}$

I need to investigate the series (Hence, when the series converges and when the series converges absolutely depending on $\alpha$). $$\sum\limits_{n\ge2}{\frac{(-1)^n}{n^\alpha \ln n}}$$ For ...
1
vote
1answer
30 views

evaluating a sum using Cauchy condensation test

Let $$\sum\limits_{n\ge1}{\frac{(-1)^n}{n^\alpha \ln n}}$$ I want to check if the sum is converges absolutely. Hence, we need to check the convergence of $$\sum\limits_{n\ge1}{\frac{1}{n^\alpha \ln ...
5
votes
2answers
160 views

What is the closed form for $S=\displaystyle\sum_{n=1}^{\infty} \dfrac{\sin ({n})}{n!}$?

How do we find the following sum (closed form)? $$S=\displaystyle\sum_{n=1}^{\infty} \dfrac{\sin ({n})}{n!}$$ Edit What about $$S=2\left(\displaystyle\sum_{n=1}^{\infty} \dfrac{\sin ...
0
votes
2answers
57 views

How to find the value of this summation?

How to solve this summation? $$S=\displaystyle\sum_{n=1}^{\infty}\dfrac{(-1)^n}{n4^{4n+1}}$$ I tried it to convert it into a definite integral but wasn't successful. Help will be appreciated.
6
votes
3answers
114 views

How to find sums like $\sum_{k=0}^{39} \binom{200}{5k}$

How do I find sums like these?-- $$S=\displaystyle\sum_{k=0}^{39} \dbinom{200}{5k}$$ that is, when there is a summation of binomial coefficients, but with jumps of some terms..?
3
votes
1answer
29 views

Inequality containing finite sum.

For what value of k the following inequality holds? $\sum_{i=1}^{n}a_{i}^3<k|\sqrt{\sum_{i=1}^{n}a_{i}}|$ I don't have any idea to solve this.
4
votes
3answers
476 views

Does a closed form exist for this summation?

How do I calculate $$\sum_{k=1}^\infty \frac{k\sin(kx)}{1+k^2}$$ for $0<x<2\pi$? Wolfram alpha can't calculate it, but the sum surely converges.
2
votes
3answers
42 views

Series question with logarithms

I want to know how to check the divergence of following sum: $\sum_{k=0}^\infty \frac{1}{\sqrt[n]{\log n}}$ I tried to use this result: $ \lim_{n \rightarrow \infty} \frac{1}{\sqrt[n]{\log n}}=1 ...
4
votes
6answers
104 views

If $\lim\limits_{x \to \infty} f(x) = 1$, can we have function $f(x)$, such that $\int_0^{\infty}f(x)dx$ converges

I know the Initiative answer, can anyone give a neat answer based on solid reasoning EDIT : $f(x)$ is continuous
0
votes
4answers
135 views
0
votes
0answers
34 views

Taking the derivative of a summation

I need to know how to solve this, I would prefer a step by step process, and not just a solution please, working with perceptrons in a neural net. n is the number of nodes in a layer, every node is ...
3
votes
1answer
59 views

Can a general formula be developed for these integrals?

Can the following integral $I$ be written wrt $n$? $$I=\displaystyle\int_{0}^{\infty} \dfrac{dx}{\sum_{k=0}^{n} x^k}$$ I found the values for $n=1,2,3$ but can we generalize it for an arbitrary ...
2
votes
4answers
115 views

Fastest way to integrate $\int_0^1 x^{2}\sqrt{x+x^2} \hspace{2mm}dx $

This integral looks simple, but it appears that its not so. All Ideas are welcome, no Idea is bad, it may not work in this problem, but may be useful in some other case some other day ! :)
2
votes
1answer
77 views

What is the inverse function of $\int{ \frac{1}{{\sqrt{x+1}}{x^n}} dx}$?

I am trying to solve $$ \frac{dy}{dt} = \alpha ((y+1)^2 - \gamma)^n \hspace{2cm} y(0)=0 $$ Here $y$ is a real-valued, monotonically increasing, positive definite function of $t$ in the interval ...
1
vote
3answers
61 views

Can you show that the LHS equals the RHS in this equation, by showing how I can get the expression on the RHS?

$$ \frac{1^2+2^2+...+(n-1)^2}{n^3} = \frac{(n-1)n(2n-1)}{6n^3} $$ Can someone show me step by step how I can transform the LHS to the RHS? If possible, using high school-level math. I have now ...
1
vote
1answer
45 views

Is an integral without a differential component on a finite number of points just a sum?

Is an integral $$\int_{\lbrace 1, 2, 3 \rbrace} f(x)$$ simply the sum $$\sum_{x=1,2,3} f(x)?$$ I ask this question because of the generalization to multiple dimensions of integration by parts ...
-1
votes
1answer
66 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.
5
votes
1answer
62 views

How many decimal representations are possible for the number 1

I know that there at least two $0.\overline{9}$ and 1 Is there a neat and more concrete way to understand this problem.
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 !
1
vote
3answers
63 views

Find all values of $c$ for which the following series converges $\sum_{n=1}^{\infty} \left(\dfrac{c}{n}-\dfrac{1}{n+1}\right)$

I know that the series in question converges when $c=1$, but I have no concrete way to find all such values of $c$ for which this is true.
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
9
votes
3answers
282 views

Proving that $ \displaystyle \gamma = \int_{0}^{1} \!\!\int_{0}^{1} \!\frac{x - 1}{(1 - x y) \log(x y)} \, \mathrm{d}{x} \, \mathrm{d}{y} $.

In 2005, J. Sondow found a surprising formula for the Euler-Mascheroni constant $ \gamma $. The formula is $$ \gamma = \int_{0}^{1} \int_{0}^{1} \frac{x - 1}{(1 - x y) \log(x y)} ~ \mathrm{d}{x} ~ ...
4
votes
3answers
164 views

Infinite Sum of algebraic expression

Prove that $$\sum_{i=1}^{\infty} \frac{1}{i(2i+3)} = \frac89 -\frac23\ln2$$ I tried using integration but failed miserably. Hints please.
2
votes
0answers
34 views

statements about summation

Could you help me prove this statements about summation? I know that the second prove is easy of be written, but can I put that summation before cos(theta) and sin(theta)? yes. But why? Do you ...
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 ...
12
votes
4answers
245 views

Show $\sum_{k=1}^{\infty}\left(\frac{1+\sin(k)}{2}\right)^k$ diverges

Show$$\sum_{k=1}^{\infty}\left(\frac{1+\sin(k)}{2}\right)^k$$diverges. Just going down the list, the following tests don't work (or I failed at using them correctly) because: $\lim ...
1
vote
4answers
93 views

Prove the sum $\sum_{n=1}^\infty \frac{\arctan{n}}{n}$ diverges.

I must prove, that sum diverges, but... $$\sum_{n=1}^\infty \frac{\arctan{n}}{n}$$ $$\lim_{n \to \infty} \frac{\arctan{n}}{n} = \frac{\pi/2}{\infty} = 0$$ $$\lim_{n \to \infty} \frac{ ...
1
vote
1answer
42 views

Prove that $\sum_{k=-\infty}^\infty e^{-j2\pi f k T}=\sum_{k=-\infty}^\infty\delta(f-\frac{k}{T})$

This is part of a proof itself. $\sum_{k=-\infty}^\infty e^{-j2\pi f k T}=\sum_{k=-\infty}^\infty\delta(f-\frac{k}{T})$ $\delta$ is Dirac function. It's been a while I am thinking about this part ...
0
votes
1answer
54 views

Find the set of convergence $\sum_{n=1}^{\infty} \frac{1+x^n}{1-x^n}$

How the interval [a, b]: $x \in [a,b]$ can be found for the next sum? $$\sum_{n=1}^{\infty} \frac{1+x^n}{1-x^n}$$ The sence to check the next limit $$\lim_{n \to \infty} \frac{1+x^n}{1-x^n} = ...
1
vote
1answer
38 views

Can we possibly exchange summation and integration with negative values?

This is an attempt to go further than this answer. Essentially, we have either a summation of an integral: $$\sum_x{ \left( \int{ f(x)dx } \right) } \tag{1}$$ ...or an integral of a summation: ...
2
votes
1answer
63 views

When can we use substitution for both integrals and summations?

This question is partially inspired by Qiaochu Yuan's answer to "Will moving differentiation from inside, to outside an integral, change the result?". Essentially, I would like to know, if we have: ...
0
votes
0answers
12 views

Determining the parameters of a limit equation

Let an=(3n^3+2n^2+n+10)^(1/3) -an-b . Let A { (a,b)∈R2 | lim as n->infinity of an=1/9^(1/3) }. And I'm supposed to find S=Σ(a^3+27b^3) . My attempt : I've worked on the limit and got to this point ...
1
vote
3answers
37 views

Finding an exact solution to a difference equation

How would I solve an equation of the form: $u(n+1)=1/2u(n)+(1/3)^n$ when $u(0)=1$? I got an answer of the form $u(n)= c + \sum(1/3)^j*2^{j-1}$ but believe this is incorrect?
8
votes
2answers
214 views

How prove this sum $1+\sum_{n=1}^{\infty}(1+x^n)(\frac{(1-y)(1-yx)(1-yx^2)\cdots(1-yx^{n-1})}{(y-x)(y-x^2)(y-x^3)\cdots(y-x^n)}=0$

let $|x|<1,|y|>1$, show that $$1+\sum_{n=1}^{\infty}\left((1+x^n)\left(\dfrac{(1-y)(1-yx)(1-yx^2)\cdots(1-yx^{n-1})}{(y-x)(y-x^2)(y-x^3)\cdots(y-x^n)}\right)\right)=0$$ by this sum,I ...
1
vote
4answers
87 views

Finding the sum of series $\displaystyle\sum \limits_{n=1}^{\infty} (-1)^{n}\frac{n^2}{2^n}$

I have some problems in finding the values of series that follow this pattern: $$\sum \limits_{n=0}^{\infty} (-1)^{n}*..$$ For example: I have to find the value of this series $$\sum ...
0
votes
1answer
18 views

Directional Derivate for sums

I know the directional derivative as $D_uf(x)=\nabla f(x) . u$ But I do not know how this applies here?
1
vote
1answer
34 views

How to evaluate a limit that involves matrices

I've stumbled upon this problem while I was browsing through the contents of an admission exam . I've struggled tremendously with this exercise and I've got no idea what do to next , it's eating me ...
1
vote
1answer
74 views

Why does a sum of a series belong to Calculus?

When I use Microsoft Mathematics I see the sum of a series belonging to Calculus inside of the calculator pad. Can someone explain, why does the sum of a series belong to Calculus?
3
votes
1answer
57 views

Evaluate the sum $1! + 2! + 3! + \cdots + n! \le ?? < (2n)!$

How to evaluate the sum below as close as possible? $$1! + 2! + 3! + \cdots + n! \le ??? $$ Is the next evaluation $ 1! + 2! + 3! + \cdots + n! \le n n! < (2n)! $ correct?
2
votes
2answers
38 views

How to find if this series is convergent or divergent

I have to find if this series is convergent or divergent. This is the series: $\sum_{n=1}^\infty{\frac{sin(5n)}{5^n}}$ I can't use the Ratio Test, and I don't know what to do with the sine in the ...
0
votes
0answers
32 views

Summing gamma functions

Suppose I'm practicing penalties. Initially my goalscoring probability is $p$. As I miss a chance, the probability increases by constant value $q$. So the probability that I score my first goal at ...
2
votes
1answer
35 views

Numerical sum problem.

I just started the series chapter and I come across some series that I don't know how should I resolve them. All of them have the same structure : $$ \sum_{n=0}^{\infty} (-1)^n ... $$ For example: $$ ...
4
votes
0answers
137 views

Integral $I=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx$

Hi I am trying to integrate and obtain a closed form result for $$ I:=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx. $$ Here is what I tried (but I do not think this is ...
0
votes
2answers
46 views

Intervel of Convergence of a Power Series

Can anyone explain how to do this problem? I think you might be able to approach it with the ration test but I'm unsure. Any help is greatly appreciated! $$\sum_{n=0}^{\infty} \frac{(2x-3)^n}{n \ ...
2
votes
1answer
80 views

$\sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}$

Hi I am trying to calculate the sum given by $$ \sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}=\ = \sqrt{\frac{\pi}{\alpha}} e^{\beta^2/(4\alpha)} ...