All Questions
8
questions
0
votes
0
answers
67
views
Is it possible to rewrite this sum in terms of some power series?
Is it possible to rewrite this sum in terms of some power series, maybe some cosine power series?
$$\sum_{n=0}^{\infty} \dfrac{x^{2n}}{2^{2n}(n!)^2}$$
- 181
1
vote
3
answers
1k
views
it's possible to invert summation/ series limits?
If the summation just sum every term i was thinking that for instance 1+2+3+4 = 4+3+2+1
so why this $$\sum\limits_{i=1}^{n} (3i)\ = \sum\limits_{i=n}^{1} (3i) $$ is not true ?
And how i can ...
- 241
0
votes
1
answer
49
views
Theoretical Procedure for Power Series Equation:
If I have the following equation: \begin{equation}2c_0(x-1)+\sum_{k=2}^\infty[(c_{k-2}+2c_{k-1})(x-1)^k]+\sum_{k=0}^\infty[(c_{k+2}(k+2)(k+1)+kc_k+(k+1)c_{k+1}+c_k)(x-1)^k]=0 \end{equation}
I was ...
- 2,638
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(...
- 2,638
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
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.
$$\frac{1}{3}\sum_{i=1}^{\lfloor x^2 \rfloor} {\Biggl(\Bigl(\sqrt{(i-1)}\Bigr)(...
- 162
4
votes
5
answers
161
views
Find the sum of $1-\frac17+\frac19-\frac1{15}+\frac1{17}-\frac1{23}+\frac1{25}-\dots$
Find the sum of $$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
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 ...
