All Questions
Tagged with summation power-series
362
questions
2
votes
1
answer
347
views
sum powers with constant exponent
I'd like to know if there is a solution to the following equation and how to solve it, thank you.
$$a^t + b^t + \cdots + z^t = Q$$
I want to find $t$, knowing $a,b,\cdots,z$ and $Q$
(Also is there ...
- 23
0
votes
1
answer
89
views
Series Summation
Consider the given sum.
$$\sum^{\infty}_{n=0} \frac{x^n}{[(2n+1)!]^3} = ?$$ Does there exist a closed form of the above summation. What is the general procedure to perform such a sum ? I tried ...
- 157
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
0
votes
1
answer
1k
views
Taking derivatives of a power series
I've been working on understanding power series, and came across a problem asking for the derivative of a certain power series and for the derivative to be a summation with a lower limit equal to zero....
2
votes
4
answers
94
views
Why is the inequality $\sum_{n=1}^{\infty} \frac{1}{n^2} \leq 1 + \int_1^{\infty} \frac{1}{x^2}$ true?
$$\sum_{n=1}^{\infty} \frac{1}{n^2} \leq 1 + \int_1^{\infty} \frac{1}{x^2}dx$$
I'm having trouble figuring out why the inequality above is true. I understand the following inequality:
$$\int_1^{\...
- 461
4
votes
2
answers
299
views
Transfer between integrals and infinite sums
So I was watching a video on YouTube about how $$\sum_{i=1}^\infty \frac{\chi(i)}{i} = \frac{\pi}{4}$$ (note that $\chi(i) = 0$ for even numbers $i$, $1$ for $\text{mod}(i, 4) = 1$, and $-1$ for $\...
- 2,402
1
vote
3
answers
94
views
For what $x>0$ series $\sum_{n=1}^{\infty} x^{1+\frac{1}{\sqrt2}+\frac{1}{\sqrt3}...+\frac{1}{\sqrt n}}$ is convergent?
For what $x>0$ series $\sum_{n=1}^{\infty} x^{1+\frac{1}{\sqrt2}+\frac{1}{\sqrt3}...+\frac{1}{\sqrt n}}$ is convergent?
By logarithmic test,
$$ \lim_{n\rightarrow \infty}\left(n \log\frac{u_n}{u_{...
- 1,614
4
votes
5
answers
161
views
Find the sum of $1-\frac17+\frac19-\frac1{15}+\frac1{17}-\frac1{23}+\frac1{25}-\dots$
Find the sum of $$1-\frac17+\frac19-\frac1{15}+\frac1{17}-\frac1{23}+\frac1{25}-\dots$$
a) $\dfrac{\pi}8(\sqrt2-1)$
b) $\dfrac{\pi}4(\sqrt2-1)$
c) $\dfrac{\pi}8(\sqrt2+1)$
d) $\dfrac{\pi}4(\sqrt2+1)$
...
- 45
0
votes
4
answers
74
views
Show that $\sum_{n=1}^{\infty} nx^n/(n-1)! = e^xx(x+1)$
Please excuse my relatively novice skills, I'm first year (of 5) on my masters in mathematics.
I'm trying to show that
$$
\sum_{n=1}^{\infty} \frac{nx^n}{(n-1)!} = e^xx(x+1), \forall x.
$$
I already ...
- 19
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:
$$\int_{0}^{1} \frac{t^2}{1-tx} dt = \...
- 41
1
vote
1
answer
1k
views
Finding the first 3 terms of a power series... I am confused as to whether I can just plug numbers in or if there is more.
The power series $J_0$= $$\sum_{n=0}^\infty\frac{(-1)^{n}x^{2n}}{(n!)^22^{2n}}$$
Ive plugged in n=0,1,2 and have gotten 1-$\frac{x^2}{4}$+$\frac{x^4}{64}$
Is this all that is required?
The ...
- 151
1
vote
2
answers
71
views
Finding the sum $\sum_{n=0}^\infty\frac{(x+1)^{n+2}}{(n+2)!}$. I cannot use my usual methods that I am use to.
I am asked to find the sum of the series $$\sum_{n=0}^\infty\frac{(x+1)^{n+2}}{(n+2)!}$$
For some reason (that I don't understand) I can't apply the techniques for finding the sum of the series that ...
- 151
0
votes
1
answer
228
views
I have found a Taylor series and a Maclaurin series for a function about x=0.... I have a couple of questions.
The first part of my question (for my homework) states "Find the Taylor series about $x=0$ of the function $f(x)=\frac{1}{(1-x)^2}$ "...
I have found the taylor series of $\frac{1}{(1-x)^2}$ to be $...
- 151
3
votes
0
answers
71
views
Computing closed form of $\sum_{k=0}^n b^{\alpha^k}$
Let $$\sum_{k=0}^n b^{\alpha^{k}}$$ be a sum where $b \in \mathbb N_+$ and $\alpha \in \mathbb R, \alpha > 1$. What is its name and how can I calculate its closed form?
\begin{align}
\sum_{k=0}^n ...
- 131
3
votes
0
answers
85
views
Replacing $n!$ with Stirling's approximation in $e^x = \sum_n \frac{x^n}{n!}$
I was wondering if there is a closed-form expression for
$$\sum_{n=0}^{\infty} \frac{x^n}{e^{-n}n^n},$$
although I expect there is none because Mathematica cannot compute it. However, from Stirling'...
- 778
1
vote
2
answers
89
views
Study simple convergence of $\sum_{n=0}^{\infty} x^{2n}$ on $[0,1[$
I have to study
1 ) the simple convergence of
$$S(x) = \sum_{n=0}^{\infty} x^{2n}$$
and
2) the uniform convergence
My attempts :
1)
$\forall x \in [0,1[$ $$S_n(x) = \frac{1-(x^2)^{n+1}}{1-x^2}...
- 239
4
votes
1
answer
180
views
Evaluating an infinite sum q-series
I have the following expression
$$ \sum_{n=-\infty}^{+\infty} \frac{1}{(1-q^{2n-1})(1-q^{m-2n})}$$
where $m$ is an odd positive integer and $|q|<1$.
Given this, the sum should converge to a ...
- 334
5
votes
0
answers
113
views
On the sum of Double Power Series $\sum\limits_{-\infty}^{+\infty}\space\frac{2^n}{{x}^{2^n}+1}$ [duplicate]
Let $\,{x\in\mathbb{C}}\,$, show that:
$$ S_{\small-}=\sum_{n=1}^{\infty}\space\frac{2^{-n}}{{x}^{2^{-n}}+1}\,=\frac{1}{\log{x}}-\frac{1}{x-1}\qquad\qquad\qquad\tag{1} $$
$$ S_{\small+}=\sum_{n=...
- 3,690
10
votes
4
answers
975
views
Prove $\sum_{i=0}^n (-1)^{n-i} \binom{n+1}{i} (i+1)^n = (n+2)^n$
I found through simulations that
$$\sum_{i=0}^n (-1)^{n-i} \binom{n+1}{i} (i+1)^n = (n+2)^n$$
Is there any proof of this? I've tried to solve it by:
Induction, but it gets too messy.
Binomial ...
- 12.1k
0
votes
1
answer
728
views
Expansion of summation of power series raised to a power
I need to simplify the expression written below to get the $ x $ term in its simplest form:
$$
\ E=\left(\sum_{k=1}^b \sum_{n=0}^\infty Z_n(a,k) x^\frac{n+k}{2} \right)^t ,\
$$
where
$$
\ Z_n(a,k)=\...
- 53
-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.
$$\sum_{i=1}^{n-1} \...
2
votes
3
answers
470
views
Changing of index in sum notation
The following is the derivative of the power-series expansion of $e^x$. I can't seem to understand why the starting point changes to $1$ after the second $\text{“}{=}\text{''}.$ Would this surely not ...
user571032
0
votes
0
answers
62
views
Finding the sum of $\sum_{i=0}^{n}r^{ai^2+bi+c}$
I'm trying to find a closed form for $\sum_{i=0}^{n}r^{ai^2+bi+c}$.
I'm thinking I might be able to use the geometric sum for a polynomial of one less degree, $\sum_{i=0}^{n}r^{ai+b} = \frac{r^{b}}{...
- 293
2
votes
2
answers
56
views
Find the derivative of a function that contains a sum
If $$f(x)=\sum_{n=1}^\infty{\frac{{(-1)}^{n+1}}{n\cdot3^n}{(x-3)}^n}$$ find $f''(-2)$.
I know that, by ratio test, the previous sum converges iff $x\in(0,6)$.
I did:
$$\begin{matrix}
f'(x)&=&...
- 2,269
-1
votes
1
answer
71
views
Closed form for a series involving exponential functions
Is there a closed form for the series $\sum_{n=0}^{\infty} \frac{(-1)^n}{n!}\mathrm{e}^{-n \theta}$ ?
The series is convergent for all real values of $\theta$ as $\lim_{n \to \infty} \frac{\mathrm{e}...
- 145
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
2
votes
1
answer
67
views
summation notation rules
I have a summation series as following:
$$\sum_{n=1}^N f(n)=-\sum_{n=1}^Ng(n)+\frac{X}{4}-\sum_{i=1}^3\bigg|\sum_{n=1}^N h_i(n)\bigg|^2$$
where $N$ is some known integer and $X$ is a constant.
...
- 219
1
vote
2
answers
67
views
For which $x\in \mathbb{R},$ the series $\sum_{n=1}^\infty\frac{\cos^{2n}x} n$ is differentiable?
Let $\displaystyle\sum_{n=1}^\infty \frac{\cos^{2n}x}{n}$. For which $x\in
\mathbb{R},$ is the series differentiable?
My attempt:
I know the series converges pointwise for every $x\ne \pi\cdot k$, ...
- 507
0
votes
1
answer
55
views
Find the primitive function of $\sum_{n=1}^{\infty}\frac{(-1)^n(2n+2)}{n!}(x-1)^{2n+1}$
Find the primitive function of
$\sum_{n=1}^{\infty}\frac{(-1)^n(2n+2)}{n!}(x-1)^{2n+1}$
My attempt:
In order to integrate, I'm trying to find the radius of convergence:
Let $t=(x-1) \Rightarrow \...
- 507
0
votes
1
answer
1k
views
Summation of infinite exponential series
How is the given summation containing exponential function
$\sum_{a=0}^{\infty} \frac{a+2} {2(a+1)} X \frac{(a+1){(\lambda X)}^a e^{-\lambda X}}{a! (1+\lambda X)}=\frac{X}{2} (1+ \frac{1}{1+\lambda X})...
- 93
0
votes
0
answers
95
views
Closed form for $Q_{m}(x)=\sum _{n=0}^{\infty } \frac{ (-1)^n}{n+m}\frac{x^{n}}{n!}$
I've just stumbled upon this
$$
Q_{m}(x)=\sum _{n=0}^{\infty } \frac{ (-1)^n}{n+m}\frac{x^{n}}{n!}
$$
and i'd like to know if it has a closed form.
Note that $m \geq1 $ is an integer.
Thanks.
- 5,627
1
vote
1
answer
495
views
Sum of the first n terms of series $\frac{x^{3n}}{3n(3n-1)(3n-2)}$
I need to find sum of first n terms of series $\sum_{1}^{\infty} {\frac{x^{3n}}{3n(3n-1)(3n-2)}}$. I tried but I just don't know how to transform it into any known form of power series.
EDIT: I tried ...
- 317
2
votes
1
answer
209
views
Bound for a double sum and power series
I am looking for a bound for the sum of of the following form:
$$\sum_{1 \leq l <k \leq n} (k-l)^{\gamma}$$
and $\gamma >1$.
Does somebody knows what would be an upper bound for such sum ...
- 91
3
votes
3
answers
216
views
Evaluate the sum using derivatives and generating functions
Evaluate $\sum_{k=1}^{n-3}\frac{(k+3)!}{(k-1)!}$.
My strategy is defining a generating function,
$$g(x) = \frac{1}{1-x} = 1 + x + x^2...$$
then shifting it so that we get,
$$f(x)=x^4g(x) = \frac{x^4}...
- 199
1
vote
1
answer
62
views
Limit value using serie definition
I need to find the limit value of $$\lim_{x\to\:0}\frac{sin(\frac{1}x)}{\frac{1}x}$$
I wanted to do it with the serie definition of sinus and I come to the result:
$$ 1 -\lim_{x\to\:0} \sum_{i=0}^{\...
- 119
1
vote
3
answers
4k
views
Deriving formula for partial sum of power series [duplicate]
I have the sum $$\sum_{i=0}^n i\cdot a^i$$ and according to wolframalpha, the partial sum evaluates to
$$ \frac{a (n\ a^{n + 1} - (n + 1) a^n + 1)}{(a - 1)^2}.$$
How does one arrive at the above ...
- 45
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
12
votes
5
answers
301
views
Sums of $5$th and $7$th powers of natural numbers: $\sum\limits_{i=1}^n i^5+i^7=2\left( \sum\limits_{i=1}^ni\right)^4$?
Consider the following:
$$(1^5+2^5)+(1^7+2^7)=2(1+2)^4$$
$$(1^5+2^5+3^5)+(1^7+2^7+3^7)=2(1+2+3)^4$$
$$(1^5+2^5+3^5+4^5)+(1^7+2^7+3^7+4^7)=2(1+2+3+4)^4$$
In General is it true for further increase ...
- 13.2k
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
1
vote
2
answers
196
views
Closed-form summation of $\sum_{i=1}^n i\frac{x^i}{i!}$
Is there any way to find the closed-form of this finite summation, knowing that x<1? It is part of a larger equation that I am trying to solve/simplify, which has proven to use a lot of theory that ...
2
votes
1
answer
278
views
Deriving a formula for the coefficients of a power series.
Here is the image containing the question
Consider the two truncated power series $u(t)=u_0+u_1x+u_2x^2+u_3x^3+u_4x^4+\cdots+u_nx^n$ and $d(t)=d+d_1x+d_2x^2+d_3x^3+d_4x^4+\cdots+dx^n$. Show ...
- 125
-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
5
votes
2
answers
598
views
Generalization of Faulhaber's formula
Is there a way to calculate a sum of non-integer positive powers of integers?
$\sum_{k=1}^nk^p: n \in \mathbb{N}, p \in \mathbb{R^+}$
There's a Faulhaber's formula, but as far as I can see, it is ...
- 230
1
vote
0
answers
74
views
How to find the summation of the following series?
I want to find
$$
\sum_{n=0}^\infty r^{n^2} \quad \text{and} \quad \sum_{n=0}^\infty n^2 r^{n^2}
$$
r<1
0
votes
1
answer
1k
views
How to find sum of the power series $\sum_{n=1}^{\infty} x^2 e^{-nx}$?
I got this power series $$\sum_{n=1}^{\infty} x^2 e^{-nx} $$
And i need to prove uniform convergence when $x \in[0;1]$ and find this sum. I have proven uniform convergence, but i have no idea how to ...
- 535
1
vote
2
answers
217
views
Sophomore's Dream : integral not defined in x=0
Sophomore's dream is the identity that states
\begin{equation}
\int_0^1 x^x dx = \sum\limits_{n=1}^\infty (-1)^{n+1}n^{-n}
\end{equation}
The proof is found using the series expansion for $e^{-x\...
- 11
1
vote
1
answer
300
views
Proof that a sum is monotonically decreasing
This question is a follow up of the question asked in: Sum of a sequence which is neither arithmetic nor geometric
I have the following sum which doesn't seem to have a closed-form expression:
$$S_n =...
2
votes
1
answer
96
views
Solving ODE with power series
In some old notes, I found an exercise in which it was asked to solve this ODE, in a neighbourhood of $x_0=1$:
$$xy''(x) - 3y(x) = 2x^2$$
I tried to solve it but I'm getting stuck. Let me show you ...
- 341
10
votes
2
answers
2k
views
Find the value $\binom {n}{0} + \binom{n}{4} + \binom{n}{8} + \cdots $, where $n$ is a positive integer.
Given
$$(1+x)^n= \binom {n}{0} + \binom{n}{1} x+ \binom{n}{2} x^2+ \cdots + \binom {n}{n} x^n.$$
Find the value $\binom {n}{0} + \binom{n}{4} + \binom{n}{8} + \cdots $, where $n$ is a positive ...
- 8,912