All Questions
8
questions
1
vote
1
answer
60
views
Why does $f^{(k)}(0)$ exists in Rudin's PMA Corollary 8.1?
is plugging $0$ in (6) result to $0^0$?
here is conditions of $8.1$
1
vote
1
answer
531
views
Calculating sum of series using derivative of a function
We're given the following problem:
"We know that $\frac{1}{1 - x} = \sum_{k=0}^{\infty} x^k $ for $ -1 < x < 1 $. Using the derivative with respect to $x$, calculate the sum of the following ...
- 181
0
votes
2
answers
62
views
Show that a the sum function of a power series is differentiable twice and that $f''(x) = \frac{1}{49-7x}$
I am studying for my analysis exam and are to consider the power series
$$
\sum_{n=2}^\infty \frac{1}{n(n-1)7^n}z^n
$$
with the sum function for $x \in ]-7,7[$ given by
$$
f(x) = \sum_{n=2}^\infty \...
- 917
0
votes
1
answer
1k
views
Taking derivatives of a power series
I've been working on understanding power series, and came across a problem asking for the derivative of a certain power series and for the derivative to be a summation with a lower limit equal to zero....
2
votes
2
answers
56
views
Find the derivative of a function that contains a sum
If $$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
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
3
votes
2
answers
144
views
Summation of infinite series
If we know the series sum given below converges to a value $C$(constant)
$$\sum_{n=0}^{\infty}a_n =C \tag 2$$ Can we generate following in terms of C. values of $a_n$ will tend to zero as n goes to ...
- 1,717
6
votes
1
answer
164
views
Simplify $\sum_{n=0}^{N}\binom{N}{n} \frac{a^{N-n}}{n!} \frac{d^n}{dx^n} f(x)$
Simplify the following expression
$$S_N = \sum_{n=0}^{N}\binom{N}{n} \frac{a^{N-n}}{n!} \frac{d^n}{dx^n} f(x), $$
where $a$ is a real number and $f(x)$ is an analytic real function.
What is $\lim_n ...
- 1,258