All Questions
41
questions
1
vote
1
answer
60
views
Why does $f^{(k)}(0)$ exists in Rudin's PMA Corollary 8.1?
is plugging $0$ in (6) result to $0^0$?
here is conditions of $8.1$
4
votes
1
answer
89
views
A conjecture involving series with zeta function
Recently, I tried to evaluate a limit proposed by MSE user Black Emperor. In the process of evaluating the limit, I have obtained the following equality.
$$
\lim_{N\rightarrow \infty} \sum_{n=0}^{N-2}{...
- 1,250
0
votes
0
answers
60
views
Rewriting a sum with a floor function as upper limit
I am having some trouble in rewriting a sum whose upper limit is given in terms of a floor function $\lfloor \cdot \rfloor$. The task is to prove that both sides of the following expression coincide:
$...
- 91
0
votes
0
answers
105
views
Manipulation with the following infinite sum
Calculating some observable, I obtained the following-like converges sum
$$
S = \sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\sum_{p=0}^{\min(n,k)} \frac{x^n}{n!} \frac{y^k}{k!} F(p),
$$
where $F$ - some ...
- 121
0
votes
0
answers
65
views
How to find the sum of a power series without knowing the actual power series
How do I find the sum of this series?
$$ \sum_{n=1}^{\infty}\frac{(-1)^{n-1}n}{(n+1)2^{n-1}} $$
The approach I wanted to use is to find a power series that can become this number series for a certain ...
- 11
0
votes
1
answer
71
views
Evaluate $\sum_{i=2}^{\infty}{\frac{n}{n^2-1}}x^n$
Evaluate $$\sum_{i=2}^{\infty}{\frac{n}{n^2-1}}x^n$$ using the fact that
$${\frac{n}{n^2-1}} = {\frac{1}{2(n-1)}} + {\frac{1}{2(n+1)}}$$
So far I have proven that the Radius of Convergence is 1 and ...
- 63
0
votes
2
answers
52
views
What is the Radius of Convergene of $f(z) = \sum_{n=0}^\infty \frac1{4^n}z^{2n+1}$
Here's what I have:
$f(z) = z + \frac14z^3+\frac1{4^2}z^5+...$
So, my coefficients are either $0$ or $\frac1{4^n}$ with $\frac1{4^n}$ being the supremum.
So, $\limsup \limits_{n \to \infty} |c_n|^{\...
- 70
10
votes
2
answers
2k
views
Find the sum: $\sum_{n=0}^\infty \frac{(n!)^2}{(2n)!}x^n$ [duplicate]
Find the sum:
$$\sum_{n=0}^\infty \frac{(n!)^2}{(2n)!}x^n$$
My try:
I played a bit with the coefficient to make it look easier/familiar:
First attempt:
$$\begin{align}
\sum_{n=0}^\infty \frac{(n!)^2}{...
- 877
1
vote
1
answer
49
views
Show that $F(x)=\frac{x}{(1-x)^2}-\frac{2x}{(2-x)^2}=\sum_{n=0}^{\infty}n(1-2^{-n})x^n$
I've been working on a recent exercise question where I was asked to show that:
$$F(x)=\frac{x}{(1-x)^2}-\frac{2x}{(2-x)^2}=\sum_{n=0}^{\infty}n(1-2^{-n})x^n$$
Now I cansee that the infinite sum is ...
- 115
1
vote
4
answers
111
views
Evaluating the sum $\sum\limits_{n=1}^{\infty}\frac{x^{2n}}{(2n+1)!}$
I'm having difficulties with the sum above. My first attempt was to rewrite it like
$$
\sum\limits_{n=1}^{\infty}\frac{x^{2n}}{(2n+1)!}=\frac{1}{x}\sum\limits_{n=1}^{\infty}\frac{x^{2n+1}}{(2n+1)!}
$$
...
- 81
0
votes
1
answer
46
views
Sum of a finite series almost like gp
Let $a>1$ and consider the following finite series:
$$
1+\frac{2}{a}+\frac{3}{a^2}+\cdots+\frac{n}{a^{n-1}},
$$
where $n\geq 1$ is a fixed quantity.
Then is the above series uniformly bounded by a ...
- 713
1
vote
3
answers
117
views
Suppose $\sum_{n=1}^{\infty}a_nx^n=f(x)$, then what can we say $\sum_{n=1}^{\infty}\frac{a_n}{n^3}x^n$ and the limit?
Let $f(x)=\frac{1}{(1-x)(1-x^4)}$, let $a_n$ be the nth term of the maclaurin expansion of $f(x)$.
What can we say about the power series $a_0+\sum_{n=1}^{\infty}\frac{a_n}{n^3}x^n$? Can we express ...
- 293
0
votes
0
answers
144
views
Radius of Convergence of the 'product' of two power series of the same radius of convergence
I have come across a problem involving the radius of convergence of power series that I can't solve for the life of me, any help would be greatly appreciated. I cannot use the lim sup method to solve ...
- 41
2
votes
2
answers
138
views
Finding the exact sum of this power series
I am currently studying power series and have come across a problem I am having difficulties with. I have done some looking around on the website for a similar problem but I cant find anything that ...
- 41
0
votes
2
answers
254
views
Infinite power series sum $\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)3^n}$
Using theorems about differentiation or integration of power series calculate infinite sum of
$$
\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)3^n}
$$
The answer should equal to $\frac{\pi}{2\sqrt3}$.
I ...
- 2,190
0
votes
1
answer
44
views
Is it possible to find a closed form $\sum_{i=0}^n x^{f(i)}$ in general? For $f : i \mapsto i + i^2$?
Let be $f : \mathbb{N} \to \mathbb{N}$, I'm interested if it is possible to find a closed form of $\displaystyle \sum_{n=0}^{p} x^{f(n)}$ for all $x \in \mathbb{C}$ for all $p \in \mathbb{N}$, also ...
- 1,920
11
votes
4
answers
2k
views
A power series $\sum_{n = 0}^\infty a_nx^n$ such that $\sum_{n=0}^\infty a_n= +\infty$ but $\lim_{x \to 1} \sum_{n = 0}^\infty a_nx^n \ne \infty$
Let's consider the power series $\sum_{n = 0}^{\infty} a_nx^n $ with radius of convergence $1$. Moreover let's suppose that : $\sum_{n = 0}^{\infty} a_n= +\infty$. Then I would like to find a sequence ...
- 666
1
vote
3
answers
94
views
For what $x>0$ series $\sum_{n=1}^{\infty} x^{1+\frac{1}{\sqrt2}+\frac{1}{\sqrt3}...+\frac{1}{\sqrt n}}$ is convergent?
For what $x>0$ series $\sum_{n=1}^{\infty} x^{1+\frac{1}{\sqrt2}+\frac{1}{\sqrt3}...+\frac{1}{\sqrt n}}$ is convergent?
By logarithmic test,
$$ \lim_{n\rightarrow \infty}\left(n \log\frac{u_n}{u_{...
- 1,614
0
votes
1
answer
39
views
Solving integrals with power series
Okay, so I'm looking at the anwers to a question where you're supposed to solve a definite integral depending on $x$. And I do not understand the equality below:
$$\int_{0}^{1} \frac{t^2}{1-tx} dt = \...
- 41
3
votes
0
answers
85
views
Replacing $n!$ with Stirling's approximation in $e^x = \sum_n \frac{x^n}{n!}$
I was wondering if there is a closed-form expression for
$$\sum_{n=0}^{\infty} \frac{x^n}{e^{-n}n^n},$$
although I expect there is none because Mathematica cannot compute it. However, from Stirling'...
- 778
-1
votes
1
answer
59
views
Evaluating Sum of $\dfrac{i}{(-x)^i}$ [duplicate]
I would like to ask if the expression below can be simplified using standard summation properties? Or should I dive into much deeper concepts like the power series? Thank you.
$$\sum_{i=1}^{n-1} \...
0
votes
2
answers
76
views
How to see that equalities follow from summation by parts?
Let $\{a_n\}$ be a sequence of real numbers. Let $s_n = a_0+...+a_n$. The following equalities appear in a proof I'm reading
\begin{align}
\sum_{n=0}^\infty a_nx^n &= a_0 + \sum_{n=1}^\infty (s_n ...
- 735
0
votes
2
answers
104
views
Find the sum of infinite series
Find the sum of infinite series
$$\frac{1}{5}+\frac{1}{3}.\frac{1}{5^3}+\frac{1}{5}.\frac{1}{5^5}+...$$
I'm trying by consider this sum as S and then multiply $\frac{1}{5}$ and substract from S ...
- 1,246
1
vote
3
answers
193
views
Computing:$\sum_{n=0}^\infty\frac{3^n}{n!(n+3)}$
I'm learning the subject Power Series and I can't figure out how to find the sum of the series $$\sum_{n=0}^\infty\frac{3^n}{n!(n+3)}$$.
I know that the power series $\sum_{n=0}^\infty\frac{3^n}{n!(n+...
- 111
3
votes
1
answer
2k
views
Find a sum of a convergent series
Let $x_n$ be a sequence that is given by the following recursive formula:
$x_{n+1} = x_n^2 - x_n +1$, where $x_1=a \gt 1$.
Find: $$\sum_{n=1}^{\infty} \frac{1}{x_n}$$
Not sure really how to ...
- 415
-2
votes
2
answers
60
views
Compute $\sum\limits_{i+j+k=n} x^{2i+j+k}$ [closed]
How to compute for all $n\geq 1$ and all $x\in (0,1)$ the following quantity?
$$\sum\limits_{i+j+k=n,\\ i,j,k\geq 0} x^{2i+j+k}.$$
- 57
1
vote
1
answer
66
views
Values for which this sum can be defined in terms of known constants in a closed form
I'm interested in the sum,
$$\sum_{n=1}^\infty\frac{\zeta(2n)\Gamma(2n)}{\Gamma(2k+2n+2)}x^{2n}$$
Otherwise written as
$$\sum_{n=1}^\infty\frac{\zeta(2n)}{(2n)(2n+1)\cdots(2n+2k+1)}x^{2n}$$
I am ...
- 3,557
0
votes
2
answers
102
views
Find the sum of the following infinite series $e^{-x}\sum_{i=0}^{\infty}\frac{i.x^i}{i!}$
Find the sum of the following infinite series $$e^{-x}\sum_{i=0}^{\infty}\dfrac{i.x^i}{i!}$$
The summation looks like an exponential series but how to tackle that?$$ 0+\frac{x}{1!}+\frac{2x^2}{2!}+......
- 586
4
votes
4
answers
187
views
What is the sum of $\sum_{k=0}^\infty \frac{(-1)^{k+1}}{2k+1}$?
What is the sum of the following expression:
$$\sum_{k=0}^\infty \frac{(-1)^{k+1}}{2k+1}$$
I know it is convergent but I cannot evaluate its sum.
- 24.5k
6
votes
1
answer
280
views
Sum of $\sum_{n \geq 1} \frac{(\ln x +1)^n}{n^n}$
I want to find the sum of the following series
$$\sum_{n \geq 1} \frac{(\ln x +1)^n}{n^n}$$
Using theorems on integration and differentiation of series. I can set $t=\ln x+1$ so that I get
$$\sum_{...
- 2,517
1
vote
1
answer
89
views
How do I evaluate $\sum_{k=1}^nk^pr^k=?$
For this entire post, we have $r\ne1$, $n\in\mathbb N$. For the first half, $p\in\mathbb N$, and at the end $p\in\mathbb Q$.
It is well known that
$$\sum_{k=1}^nr^k=\frac{1-r^{n+1}}{1-r}$$
And
$$\...
3
votes
2
answers
74
views
Finding Exact Values of Specific Infinite Series
Prove that $\Sigma_{n=1}^{\infty}(n/2^n)=2$ and that $\Sigma_{n=1}^{\infty}(n^2/2^n)=6$.
Thoughts:
I have a feeling that if someone shows me how to do one, I'll be able to figure out the other. So ...
user322548
3
votes
6
answers
269
views
Is $\sum_{n=1}^{\infty} \frac{n-1}{n^2}$ convergent or divergent?
Is $\sum_{n=1}^{\infty} \frac{n-1}{n^2}$ convergent or divergent?
I tried ratio test but didn't seem to work, and I also know that the limit goes to zero, but I can't say its convergence because then....
- 33
0
votes
2
answers
1k
views
showing that the partial sums of $ \log(j) = n\log(n) - n + \text{O}(\log(n))$
I'm trying to show that the partial sums of $\log(j) = n\log(n) - n + \text{O}(\log(n))$
I know that $$\int_1^n\log(x)dx = n\log(n) - n + 1$$
so that this number is pretty close to what I want.
Now ...
- 13
4
votes
5
answers
338
views
Power series summation [closed]
Trying to find the sum of the following infinite series:
$$ \displaystyle\sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}}{(2n-1)3^{n-1}}$$
Any ideas on how to find this sum?
- 277
0
votes
1
answer
70
views
How to calculate this sum?
Let $x_1,\cdots,x_k$ be numbers between 0 and 1. Then is it possible to get explicit expression for the following sum:$$\sum_{n_1,\cdots,n_k\geq 1} x_1^{n_1}\times C_{n_1+n_2}^{n_2}\times x_2^{n_2}\...
- 693
1
vote
1
answer
78
views
Trying to understand a power series example from Advanced Calculus by Taylor
Example 2 from 21.1 in the book, Find an expansion in powers of $x$ of the function
$$
f(x) = \int_{0}^{1} \frac{1-e^{-tx}}{t}dt
$$
and use it to calculate $f(1/2)$ approximately.
I ...
- 11
0
votes
2
answers
462
views
Proof for multiplication of two power series
Prove that $(\sum_{k=0}^\infty u^k)^2=\sum_{k=0}^\infty (k+1)u^k$ when |u|<1.
This is a proof I need for a larger proof I was doing. I am stuck on this, so I was not able to make any notable ...
- 211
4
votes
1
answer
4k
views
Exponential series is cosh(x), how to show using summation?
I want to show that $$\cosh(x) = \sum_{n=0}^{\infty} \frac{(x)^{2n}}{(2n)!}
$$
I know that $cosh(x) = \frac{exp(x)+exp(-x)}{2}$ but i cant seem to get there from the original series. I know that $$...
- 635
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
0
votes
3
answers
165
views
Why $\sum_{n=0}^{\infty}(n+1)5^nx^n=\frac{1}{(1-5x)^2}$
Why $$\sum_{n=0}^{\infty}(n+1)5^nx^n=\frac{1}{(1-5x)^2}?$$
I know that $\sum_{n=0}^{\infty}x^n=\dfrac{1}{1-x}$, so by the same token, $\sum_{n=0}^{\infty}5^nx^n=\dfrac{1}{1-5x}$.
Thus
$$
\left(\...
- 12.8k