All Questions
48
questions
1
vote
1
answer
130
views
Borel Regularization of $\sum_{n=1}^\infty \ln(n)$ [closed]
I'm trying to solve the following taylor series
$$\sum_{n=0}^\infty \frac{x^n}{n!} \ln(n+1)$$
so I can regularize the following sum
$$\sum_{n=1}^\infty \ln(n)$$
Using Borel Regularizaiton I can use ...
0
votes
2
answers
80
views
On the convergence of $\sum_{n=0}^\infty(f(x)-T_n\{f\}(x))$
To test the efficiency of the Taylor Series at approximating functions, I was wondering whether$$\sum_{n=0}^\infty(f(x)-T_n\{f\}(x))\tag{$\star$}$$converges, where $T_n\{f\}(x)$ is the degree $n$ ...
- 6,549
3
votes
0
answers
63
views
Is there any function in which the Maclaurin series evaluates to having prime numbered powers and factorials? [duplicate]
I am searching for any information or analysis regarding the functions
$$f(x)=\sum_{n=1}^{\infty}\frac{x^{p\left(n\right)}}{\left(p\left(n\right)\right)!}$$
or
$$g(x)=\sum_{n=1}^{\infty}\frac{\left(-1\...
- 41
0
votes
1
answer
144
views
Given the alternating series. . . what is the infinite sum of . . .
"What is the infinite sum of the alternating series?"
$$\sum_{n=0}^{\infty} \frac{(-1)^n \, (5 \pi)^{2n+1}}{6^{2n+1} \, (2n+1)!}$$
This problem was given to me along with the $\cos(x)$ ...
0
votes
1
answer
67
views
Can I use a 'known' Maclaurin series to find the sum of a given series if the series lower bound is not the same?
I'm having a bit of a dilemma right now and I thought I'd ask here. Pretty much, I have a list of known Maclaurin series that I can use on my exam and they greatly help me in solving series and their ...
1
vote
1
answer
88
views
Aymptotic formula/closed form for $ \sum_{r=1}^{n} {n \choose r} \frac{f^{(r-1)}(1)}{(r-1)!}$
For an $f$ infinitely differentiable on $(0,\infty)$ and real valued, consider a finite sum $$a_n= \sum_{r=1}^{n} {n \choose r} \frac{f^{(r-1)}(1)}{(r-1)!}$$
where $f^{(r-1)}(1)=\frac{d^{(r-1)}f(x)}{...
user1024284
0
votes
3
answers
190
views
Finding general term
So I was given the following prompt:
"For the following, use the definition to find the Taylor (or Maclaurin) series centered at c for the function. When writing your answers, be sure to list the ...
- 325
0
votes
2
answers
62
views
Simplified form of an exp-like sum
I notice that the following series is the taylor expansion at $x=0$ of an $e^x$ function:
$$ e^x = 1+\frac{x^1}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+ ...$$
Recently, I have the following function $f_a(x)$...
- 1,027
3
votes
0
answers
533
views
According to the "DI Method" (Tabular Method) - isn't every integral a sum of the derivatives?
Many years ago I encountered this really amazing solving technique called the "DI Method for integration" which is also called the tabular method (If I am not mistaken). It lets us choose ...
1
vote
3
answers
131
views
How to prove that $\sum _{n=0}^{\infty }\:\frac{(x^n)'}{(n-1)!} = e^{x}(x-1)$
I am trying to prove that $$\sum _{n=0}^{\infty }\:\frac{\left(x^n\right)'}{\left(n-1\right)!} = e^{x}(x+1)\tag 1$$
This sum is very similar to the derivative of exponential $(e^x)' = \sum _{n=0}^{\...
- 23
5
votes
1
answer
246
views
Evaluate: $\sum_{n=1}^{\infty} {\left(\frac{-100}{729}\right)}^n {3n \choose n}$
The questions asks to evaluate:
$$\sum_{n=1}^{\infty} {\left(\frac{-100}{729}\right)}^n {3n \choose n}$$
The answer provided is $-\frac{1}{4}$, but I don't know how to solve it. I am not sure how to ...
- 5,434
3
votes
2
answers
1k
views
Evaluate $\sum_{n=0}^{\infty} \frac{{\left(\left(n+1\right)\ln{2}\right)}^n}{2^n n!}$
Evaluate: $$\sum_{n=0}^{\infty} \frac{{\left(\left(n+1\right)\ln{2}\right)}^n}{2^n n!}$$
I am not sure where to start. The ${\left(n+1\right)}^n$ term is obnoxious as I can't split the fraction. ...
- 5,434
0
votes
2
answers
144
views
Approximate the integral $\int_0^{0.5}{x^2e^{x^2}}dx$ correct to four decimal places using a Maclaurin series.
I got $$\int_0^{0.5}{\sum_0^\infty}\frac{x^{2n+2}}{n!}dx$$ for the taylor series representation, but I'm not sure what to do next.
Do I use 0 and 0.5 as bounds for z for the Lagrange Error Bound? And ...
user738554
0
votes
1
answer
46
views
Finding the numerical approximation for the derivative $F ′ ( x )$ without the actual function
I need to find a numerical approximation for the derivative $F'(x)$ of an appropriately smooth function $F(x)$ at $x=0$. However I do not know the actual function $F$. I have this formula:
$$F'(0) \...
user635953
0
votes
0
answers
42
views
Using summation to calculate the remainder of order 3 of a multivariable function
$R_{p+1}(x) = \sum_{i_1+i_2+...+i_n = p+1} \frac{1}{i_1!i_2!...i_n!} [\frac{\partial^{p+1}f}{\partial x_1^{i_1} x_2^{i_2}...x_n^{i_n}} (c) × (x_1 - x_{0,1})^{i_1} ...(x_n - x_{0,n})^{i_n}]$
where $c \...
- 1,080