All Questions
Tagged with summation power-series
362
questions
0
votes
1
answer
26
views
Given the following summation is there a way to combine given the following orientation?
I have the following summation: \begin{equation}\sum_{n=2}^\infty c_n(n)(n-1)(x-1)^{n-2}+(x+1)\sum_{n=1}^\infty nc_n(x-1)^{n-1}+\sum_{n=1}^\infty nc_n(x-1)^{n-1}+(x+1)^2\sum_{n=0}^\infty c_n(x-1)^n +2(...
- 2,653
0
votes
0
answers
33
views
Convergence of specific power series
I have to evaluate pointwise/uniform/total convergence of this series and I didn't quite understand how to do it.
$$\sum_{k=2}^{+\infty}{\ln k \over 2+\sin k}x^k$$
For pointwise convergence: it ...
- 415
1
vote
2
answers
433
views
sigma notation- squaring the entire sum
Could someone please tell me how to expand this?
$\bigg(\sum_{i=1}^ne^{at_i-\frac{1}{2}\sigma^2t_i+\beta t_i}\bigg)^2$
i know the general formula goes something like this:
$\bigg(\sum_{i=1}^na_i\...
2
votes
1
answer
132
views
Upper bound for the sum of non-integer powers
Let $a_1, a_2, \ldots, a_k$ be a positive integers such that $a_1 + a_2 + \cdots + a_k = K$.
Is it possible to find an upper bound such that $$a_1^p + a_2^p + \cdots+ a_k^p \le f(K)$$ where $0 < p ...
- 23
1
vote
0
answers
17
views
Find the power series from the given maclaurin sequence
I have made a post regard this particular question but was incorrect in what I was asking.
The sequence needs to be written in sigma notation, not as a summation.
The given sequence is:
$$x+2x^3+x^3+...
- 1,085
1
vote
0
answers
29
views
Determine the specific value of the division of two factorial series
I want to find a specific range of $\alpha$ formula as follows.
$$\alpha =\frac{1+\frac{{{x}^{2}}}{3!}+\frac{{{x}^{4}}}{5!}++\cdots+{\frac {x^{2n}}{(2n+1)!}}}{1+\frac{{{x}^{2}}}{2!}+\frac{{{x}^{4}}}{...
- 285
3
votes
1
answer
90
views
Is there anything special about this summation I found that calculates out the square, cube, fourth, etc. power of any integer?
let me start by saying that my formatting may be way off, but it's the best I can do, and has little to do with the question, and I will make sure I am as clear as humanly possible, including showing ...
1
vote
2
answers
143
views
Double Summation indexes problem
I have the following sum:
\begin{equation}
\sum_{j=0}^{a}
\sum_{k=0}^{n-2j} c_{jk}\,\,
x^{\,j+k}
\end{equation}
Where $a=\lfloor n/2\rfloor$. I want to convert the previous sum to other like:
\...
- 199
1
vote
4
answers
111
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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
1
vote
1
answer
55
views
How can I find a simple upper bound for this sum?
I want to compute a simple upper bound for
$$S = \sum_{k=1}^{T-1} (k x^{k-1})^2$$
that depends on both $x \in [-1,1]$ and $T$.
I can compute
$$S = \frac{(-2 T^2 + 2 T + 1) x^{2 T + 2} + T^2 x^{2 ...
- 1,180
1
vote
0
answers
53
views
Find error for Taylor series
I posted this task yesterday. I got help with understanding how to solve a part of the problem but the other part still confuses me. The task is:
Use the power series to $f(x)=\frac{1}{1-x}$ to find ...
- 547
0
votes
1
answer
386
views
How to find Taylor series for $\ln(1+2x)$ using power series of $\frac{1}{1-x}$
I'm asked to find the Taylor-series to
$f(x)=\ln(1+2x)$ about $x=0$ using the power series to $\frac{1}{1-x}$. And then I must find the biggest number of $n$ necessary to estimate $\ln(1.02),$ where ...
- 547
1
vote
1
answer
38
views
Help calculating series [duplicate]
I need help with understanding how to solve this task, because I'm a bit lost at the moment.
Use the powerseries $$f(x)=\frac{1}{1-x}$$ to decide the sum of the series
$\sum_{n=1}^{\infty} n(n+1)x^...
- 547
1
vote
1
answer
89
views
Evaluating $\sum^n_{x=1}{2x-1\choose x}t^x$
Is there any technique that I can use to evaluate
$$\sum^n_{k=1}{2k-1\choose k}t^k, \quad \forall t\in\left(0,\frac{1}{4}\right)$$
It can be shown that the series converges even if $n\to \infty$ ...
- 69
0
votes
0
answers
22
views
Find the summation, given a few numbers of the sequence
If a have a sequence such as the following:
$$a_0,a_1,a_2,\cdots$$
How can I find the summation representation for it, assuming it converges?
I have searched a bit on MathExchange and found this Get ...
- 516
0
votes
1
answer
46
views
Sum of a finite series almost like gp
Let $a>1$ and consider the following finite series:
$$
1+\frac{2}{a}+\frac{3}{a^2}+\cdots+\frac{n}{a^{n-1}},
$$
where $n\geq 1$ is a fixed quantity.
Then is the above series uniformly bounded by a ...
- 713
0
votes
4
answers
241
views
How to compute $\sum_{k = 1}^{\infty} \frac{1}{k^2-\frac{1}{16}}$? [duplicate]
How do I compute $\sum_{k = 1}^{\infty} \frac{1}{k^2-\frac{1}{16}}$ ?
Mathematica says the sum converges and it somewhat looks like the Basel problem, but so far I do not know how to approach it.
- 1,804
0
votes
2
answers
75
views
Evaluate $\sum_{n=1}^\infty(-1)^{n-1}\frac{9}{4^n}.$
Evaluate $$\sum_{n=1}^\infty(-1)^{n-1}\frac{9}{4^n}.$$
My approach: using $\frac{a}{1-r}$, we have
$\frac{9}{1-\frac{1}{4}} = 12$
But the correct answer is $\frac{9}{5}$
What am I missing in ...
- 291
0
votes
1
answer
42
views
Calculate $\sum^\infty _{k=1} \frac{10^{-k-1}}{k(k+1)}$ using the Taylor series of $(1-x)\ln(1-x)$
For the Maclaurin series of $(1-x)\ln(1-x)$ I have obtained
$$\sum^\infty_{k=1}\frac{x^{k+1}}{k}-\sum^\infty_{k=1}\frac{x^k}{k}$$
I am unsure on how to use this to obtain the value of the sum: $$ \...
0
votes
1
answer
79
views
Sum of x raised to odd powers till infinity where x is less than 0 [closed]
I have an geometric series which has the following form
0.5^3 + 0.5^5 + 0.5^7 + 0.5^9 .... 0.5^inf
Is there a formula to find the sum of this series ?
- 201
-2
votes
3
answers
1k
views
How to evaluate $\sum\limits_{n=0}^\infty n^3x^n$, for $x\in[0,1)$.
How can I solve the following sum?
$$\sum_{n=0}^\infty n^3x^n$$
For $x\in[0,1)$. Do I have to try values and see if its a geometric series or something like that?
Or do I have first find the ...
0
votes
0
answers
36
views
Values of x of convergence and absolute convergence $\sum_{n=0}^{\infty} \frac{n!(2n)!}{(3n)!}x^n$
For which values of x the following power series converge and for which values of x converge absolutely: $$\sum_{n=0}^{\infty} \frac{n!(2n)!}{(3n)!}x^n$$
I found this question quite strange because I'...
- 1,326
0
votes
2
answers
97
views
Find the sum of power series
I have to find the sum of the following sequence
$$\sum_{n=1}^\infty \frac{x^n}{n-1},\quad x\geq0$$ and
How am I supposed to start?
Thanks.
- 29
0
votes
2
answers
169
views
Radius of convergence for $\sum_{n=0}^\infty n^nx^n$ and $\sum_{n=0}^\infty \frac {(-3)^n}{n}(x+1)^n$
How can one calculate the radius of convergence for the following power series:
$$\sum_{n=0}^\infty n^nx^n$$
and
$$\sum_{n=0}^\infty \frac {(-3)^n}{n}(x+1)^n$$
Regarding the first one I know ...
- 931
2
votes
2
answers
752
views
Evaluating $\sum_{k=0}^\infty\sin(kx)$ and $\sum_{k=0}^\infty\cos(kx)$
Playing with sines I wanted to find
$$
S(x)=\sum_{k=0}^\infty\sin(kx)
$$
Writing it as
$$
S(x)=\mathrm{Im}\big(A(x)\big),\quad\text{where}\quad A(x)=\sum_{k=0}^\infty e^{ikx}
$$
and using $z=e^{ix}=...
- 1,024
5
votes
1
answer
61
views
Simplification of $a_0+\sum_{n=1}^{\infty} a_n\cdot \left[\sum_{k=0}^{\infty}b_k\cdot x^k\right]^n$ where I know the expressions for all the $a_n$
I have this equation: $a_0+\sum_{n=1}^{\infty} a_n\cdot \left[\sum_{k=0}^{\infty}b_k\cdot x^k\right]^n$ where I know the expressions for all the $a_n$.
How can I simplify the multiplication of the ...
- 51
1
vote
0
answers
38
views
What does the functional power series with $\text{zero sum}$ represent in mathematics and physical world?
Suppose we have a collection of functional power series with invariant sum $0$,
i.e, assume $$F_k(x)=\sum_{n=0}^{\infty} n f_k(n,x)x^n=0, \ x \in \Bbb Q$$
where $f_k(n,x)$ is a polynomial of degree $...
- 10.8k
1
vote
1
answer
49
views
Exponent-like power series with coefficients of increasing complexity
How one deals with series like this one:
$zb + \frac{z^2}{2!}b^2(1+\frac{c}{b}) + \frac{z^3}{3!}b^3(1+\frac{2c}{b})(1+\frac{c}{b}) + \frac{z^4}{4!}b^4(1+\frac{3c}{b})(1+\frac{2c}{b})(1+\frac{c}{b})+......
- 63
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
19
votes
2
answers
758
views
Closed form of $\sum_{n = 1}^{\infty} \frac{n^{n - k}}{e^{n} \cdot n!}$
When seeing this question I noticed that
$$
\sum_{n = 1}^{\infty} \frac{n^{n - 2}}{e^{n} \cdot n!}
= \frac{1}{2}.
$$
I don't know how to show this, I tried finding a power series that matches that but ...
- 4,878
1
vote
0
answers
72
views
Why A Sum Related To The Harmonic Series Diverges [duplicate]
When I recently read about the power rule for infinite sums, I thought that the sum $$\sum_{n=1}^{\infty} \frac{1}{n^{1+\frac{1}{n}}}$$ would converge, as it is akin to a $p$-series, but with an (...
- 1,789
2
votes
1
answer
106
views
Evaluating a finite sum with square roots and simple powers. + The integral of floor(x^2) + The integral of the fractional part of x^2
I was recently integrating the floor of $x^2$ and had almost finished it, however this finite Sum was left unevaluated.
$$\frac{1}{3}\sum_{i=1}^{\lfloor x^2 \rfloor} {\Biggl(\Bigl(\sqrt{(i-1)}\Bigr)(...
- 162
1
vote
1
answer
126
views
Closed form for the series $a+a^p+(a^p)^p+\dots$
We have the arithmetic sequence in which each term is given by adding a common difference to the previous term. Then there is the geometric sequence in which each term is given by multiplying the ...
- 19.9k
1
vote
0
answers
31
views
behavior at boundary of convergence
I have a rather general question and would be happy if you could teach me how to answer it :-)
Consider for example $$f(x)=(1-x)^{1/3}$$ which one could represent as a power series: $$(1-x)^{1/3}=\...
- 9,667
0
votes
1
answer
69
views
$\sum_{n=1}^{\infty}(n \mod 7)10^{-n!}$
Is the number $$\sum_{n=1}^{\infty}(n \mod 7)10^{-n!}$$ algebraic ?
Here, $n \mod 7$ means the natural number is between 0 to 6 which is congruent to n modulo 7.
Justify your explanations, as I don'...
- 1,322
2
votes
3
answers
174
views
Sum to infinity series: $\sum_{k=0}^{\infty}\frac{(k+1)(k+3)(-1)^k}{3^k}$
Consider the following series $$\sum_{k=0}^{\infty}\frac{(k+1)(k+3)(-1)^k}{3^k}$$
I'm told to analytically find the sum to infinity and I have been given this as a clue.
$$\Sigma_{k=0}^\infty x^k = ...
- 415
-1
votes
1
answer
52
views
Summing a series exactly 1 [duplicate]
How does one go about exactly summing this series
$$\sum_{n=1}^{\infty}\frac{(-1)^nn^2}{3^n}$$
Stuck on this and not sure how to proceed. Appreciate any assistance!
0
votes
0
answers
144
views
Radius of Convergence of the 'product' of two power series of the same radius of convergence
I have come across a problem involving the radius of convergence of power series that I can't solve for the life of me, any help would be greatly appreciated. I cannot use the lim sup method to solve ...
- 41
2
votes
2
answers
138
views
Finding the exact sum of this power series
I am currently studying power series and have come across a problem I am having difficulties with. I have done some looking around on the website for a similar problem but I cant find anything that ...
- 41
0
votes
2
answers
254
views
Infinite power series sum $\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)3^n}$
Using theorems about differentiation or integration of power series calculate infinite sum of
$$
\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)3^n}
$$
The answer should equal to $\frac{\pi}{2\sqrt3}$.
I ...
- 2,190
0
votes
1
answer
132
views
Summation of Power Series
$\sum_{k=0}^n k(k − 1)(k − 2)=1/4(n+1)n(n-1)*(n-2)$
Find a general formula for $\sum_{k=0}^n k(k − 1)(k − 2) · · · (k − r + 1)$ with respect to k and r.
I know summation of power series such as $\...
- 1,913
0
votes
1
answer
44
views
Is it possible to find a closed form $\sum_{i=0}^n x^{f(i)}$ in general? For $f : i \mapsto i + i^2$?
Let be $f : \mathbb{N} \to \mathbb{N}$, I'm interested if it is possible to find a closed form of $\displaystyle \sum_{n=0}^{p} x^{f(n)}$ for all $x \in \mathbb{C}$ for all $p \in \mathbb{N}$, also ...
- 1,920
0
votes
3
answers
67
views
Prove that $\sum_{n=0}^\infty {\frac{x^{2n+1}}{(2n+1)!}} = \frac{e^x - e^{-x}}{2}$
I have been trying to show:
$\sum_{n=0}^\infty {\frac{x^{2n+1}}{(2n+1)!}} = \left(\frac{e^x - e^{-x}}{2} \right)$
I have come so far as to show:
$\begin{aligned}
\sum_{n=0}^{\infty} {\frac{x^{2n+1}}...
- 3
3
votes
2
answers
2k
views
Derive the sum of $\sum_{i=1}^n ix^{i-1}$
For the series
$$1 + 2x + 3x^2 + 4x^3 + 5x^4 + ... + nx^{n-1}+... $$
and $x \ne 1, |x| < 1$.
I need to find partial sums and finally, the sum $S_n$ of series.
Here is what I've tried:
We ...
- 157
0
votes
1
answer
99
views
Find the Taylor series of $\frac{1+z}{1-z}$ at $z_{0}=i$
I'm given the following explanation:
Let $\dfrac{1+z}{1-z} = \biggl(\dfrac{1+i}{1-i}+\dfrac{z-i}{1-i}\biggr)\biggl(1-\dfrac{z-1}{1-i}\biggr)^{-1}$ =
= $\dfrac{1+i}{1-i}\displaystyle\sum_{j=0}^{\...
- 2,338
1
vote
2
answers
58
views
In expressing arctangent as a series, why does substituting $x=1$ make sense?
arctangent can be expressed as a power series when $x$ is between $-1$ and $1$. One post argued that this was possible because the series when x=1 converges, but how do I know it converges to ...
- 11
-1
votes
2
answers
41
views
The taylor series is given determine $a_n$ for which x converges to f(x)
The taylor series of the function f(x) = $1-\mathrm{e}^{-x^2}$ around x = 0 is given by $\sum_{n=0}^{\infty} a_nx^n$
determine $a_n$ for all n $\geq$ 0 and give for which value of x the series ...
- 117
4
votes
1
answer
113
views
Compute $\sum_{n\geq1}\frac{1}{\pi^2+n^2}$ by expanding $e^{\pi x}$ in its Fourier series
Compute $$\sum_{n\geq1}\frac{1}{\pi^2+n^2}.$$
by expanding $e^{\pi x}$ in its Fourier series.
So I calculated that
$$c_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{\pi x} e^{-inx} \ dx=\frac{\sinh{(\pi^...
- 6,423
6
votes
2
answers
225
views
Does $ \sum_{i=1}^n n^{k_i} $ determines $(k_1,...,k_n)$?
Let $k_1,...,k_n\in\mathbb{N}$. Does the power sum
$$
\sum_{i=1}^n n^{k_i}
$$
uniquely determines the $n$-tuple $(k_1,...,k_n)$?
Remark: In the case $n=2$, this is true. However, when trying to ...
- 3,435
1
vote
3
answers
43
views
Clarification on alternative expression of sum
I am not familiar with alternative expression of sum shown below,
$ d_k=\sum_{i+j+l=k}a_ib_jc_l $
How it does work?
for $k = 4 $ then,
$d_{4} = \sum_{i+j+l=4}a_ib_jc_l = ...$
How do I express it ...
- 466