Tagged Questions
0
votes
0answers
14 views
Summation involving Hypergeometric exponential and factorial
I am not able to solve the following sum. Can you please provide any hints ?
$$
\sum_{y=1}^\infty {}_1F_1(1-y;2;-\pi\lambda c) \frac{\lambda^y}{y!}
$$
Thank you for your time
0
votes
0answers
17 views
Summation involving Laguerre polynomial
Is there any chance of expressing the above summation with simple elementary or special functions?
$$
\sum_{N=1}^\infty \frac {\lambda^N} {N!}\frac {1}{N}L_{N-1}^{(1)}(-\pi \lambda c)
$$
Thanks in ...
4
votes
1answer
102 views
Estimating the sum $\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$
By integral test, it is easy to see that
$$\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$$
converges. [Here $\ln(x)$ denotes the natural logarithm, and $\ln^2(x)$ stands for $(\ln(x))^2$]
I am ...
1
vote
2answers
65 views
question on summation?
Please, I need to know the proof that
$$\left(\sum_{k=0}^{\infty }\frac{n^{k+1}}{k+1}\frac{x^k}{k!}\right)\left(\sum_{\ell=0}^{\infty }B_\ell\frac{x^\ell}{\ell!}\right)=\sum_{k=0}^{\infty ...
12
votes
3answers
146 views
$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$
Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction.
$$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
0
votes
1answer
30 views
How to maximize $n!\sum^n_{k = 0}\frac{a^k(1+(-1)^{n-k})}{k!(n-k)!} \pmod{a^2}$ for a given $a$?
Short Version of the Question:
How do I maximize the value of $n!\sum^n_{k = 0}\frac{a^k(1+(-1)^{n-k})}{k!(n-k)!} \pmod{a^2}$ for a given $a$?
Long Version of the Question:
I'm currently attempting ...
0
votes
1answer
16 views
Absolute convergence.
Determine if absolutely convergent or not; Justify.
$$\sum_{n=1}^\infty (-1)^n n^2 3^{1-n} x^n \text{ s.t }|x|<3$$
if we take the abs value of $(-1)^n$ we are left with $n^{2} 3^{1-n} x^{n}$ now ...
2
votes
5answers
110 views
Proving $\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$ without induction [duplicate]
I was looking at: $$\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$$
It's pretty easy proving the above using induction, but I was wondering what is the actual way of getting this equation?
0
votes
1answer
69 views
Finding the error of the Taylor expansion of $\log(1 + x)$
The questions is as defined below.
Let $f(x)= \log(1+x)$. Show that the Taylor remainder $R_{0,k}(x)$, defined by $$R_{a,k}(x)= f(a+x) - P_{a,k}(x) = f(a+h) -\sum_{j=0}^{k} \frac ...
0
votes
1answer
42 views
Sequence of funtions question.
Kind of an odd one in my book.
The question is;
show that the Series $\sum_{1}^{\infty} \frac {1}{x^{2}-n^{2}}$ converges uniformly on any compact interval that does not contain a nonzero integer, ...
1
vote
1answer
37 views
Convergent Infinite Series
The question is deifned as Let $\sum_{0}^{\infty} a_{n}$ be a convergent series, let $\sum_{0}^{\infty} b_{n}$ be its rearangement obtained by interchangeing each even numbered term with the odd ...
3
votes
1answer
103 views
Sum involving the hypergeometric function, power and factorial functions
I am finding some trouble in calculating the following sum involving the hypergeometric function, power and factorial functions.
$$
\sum_{y=1}^\infty \mathrm{e}^z \cdot {}_1F_1\left(1-y;2;-z\right) ...
0
votes
3answers
84 views
Calculating the sum of $f(x) = \sum_{n=0}^{\infty} {n \cdot 2^n \cdot x^n}$
In Calculus, how do I calculate this sum?
$$f(x) = \sum_{n=0}^{\infty} {n \cdot 2^n \cdot x^n}$$
This is what I did so far:
$$ f(x) = 2x \cdot \sum_{n=0}^{\infty} {n \cdot 2^n \cdot x^{n-1}} $$
...
0
votes
0answers
54 views
Expected value of a Poisson sum of confluent hypergeometric functions times (version 2)
In continuation to my question here , what is the expected value of a Poisson sum of the following confluent hypergeometric function:
$$
\sum_{y=1}^{Y} (1/Y)({}_1F_1(y,1,z))
$$
where y is positive ...
3
votes
2answers
67 views
Calculating power sums using integration/derivation part by part
In Calculus, it is quite a necessity calculating the sum of a power sum of the concept: $$\sum_{n=1}^{\infty} {c_n x^n} $$
Can somebody explain how does one calculate the value of a sum using other ...
