# Tagged Questions

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### Evaluating sums and integrals using Taylor's Theorem

Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}x^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt$$ This could be used to evaluate partial sums using knowledge of the ...
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### Series Coefficient Convergence implies Uniform Convergence

Trying to find a reference for the following. Define the entire functions, $$f_n(x)=\sum_{k=0}^\infty a_{n,k}x^k\ \ \ \ \ \ \ \ \ \ \ f(x)=\sum_{k=0}^\infty a_kx^k.$$ If for each $k$, ...
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### Radius convergence of a power series…

"Suppose that $\sum_{n=0}^{+\infty}a_nx^n$ has convergence radius $R$, $R>0, \text{or }R=+\infty$. Proof that the convergence radius of $\sum_{n=0}^{+\infty}na_nx^{n-1}$ is also $R$." This seems ...
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### Conditions for convergence of $\sum\limits_{n=1}^\infty{a^nf(n)}$

assume $a>0$, and for all $n$ we have $0 \leq f(n) \leq 1$. Is there a necessary and sufficient condition on the series $f(n)$ for which $\sum\limits_{n=1}^\infty{a^nf(n)}<\infty$ ? Thanks!
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### Function as a series :

Let $f(x)=\sum_{n=0}^{+\infty}\dfrac{x^n}{n!}$. Verify that $$\int_0^xf(x)dt=f(x)-1$$ This is the exercise 3 of the section $7.4$, of Guidorizzi's Calculus, Vol. 4. What I have tried: By the ratio ...
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### $f_1 \in L^1_{loc}(\mathbb{R})$ and $f_{n+1} (x)= \int_0^x f_n(t) dt$, What is $\sum_n f_n$?

$f_1 \in L^1_{loc}(\mathbb{R})$ and $f_{n+1}(x) = \int_0^x f_n(t) dt$, What is $\sum_n f_n$? (and converges in what sense?) My attempt: Suppose $f_1$ is bounded, define the continuous linear ...
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### How does one obtain the expansion of $e^{-x^2}$ in a power series?

So I know that the Power Series $y = \displaystyle\sum_{m=0}^\infty\displaystyle\frac{(-1)^m}{m!} x^{2m}$ is equivalent to $e^{-x^2}$. Could someone show me why this is?
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### Find the radius of convergence of $\sum_{n=0}^{\infty} (3^n + (-5)^n) x^{7n}$

I have to find the radius of convergence of the series $$\sum_{n=0}^{\infty} (3^n + (-5)^n) x^{7n}$$ I know that I will have something like $|x^7|<\frac{1}{L}$. I tried finding $R$ with ...
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### Taylor Series expansion and first four terms of $7x^2 e^{-4x}$

As the series I got $$\sum_{n=0}^\infty (-1)^n(4x)^n/n!$$ which I think is right. However, I am not sure how to get the first four non zero terms.
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### Meaning of interval of convergence when approximating functions

Let's say I have a Taylor series approximation, $p(x)$, of a function $f(x)$ at $a$: $$p(x)=\sum_{n=0}^\infty{\frac{f^{(n)}(a)}{n!}(x-a)^n}$$ And that this Taylor series has a radius of ...
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### Why is the domain of convergence of a power series a perfect disk?

I've been going over power series in my Differential Equations class for approximating solutions, and one thing that's been fascinating me is the statement that there is a radius of convergence, ...
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### Calculate the value of $\sum\limits _{n=1}^{\infty }\:\dfrac{n}{2^n}$ [closed]

In a previous question it is asked to represent $f(x)=\dfrac{x}{1-x^2}$ as a power series. It gave me $\displaystyle\sum _{n=1}^{\infty \:}x\left(2x^2-x^4\right)^{n-1}$. Then they ask to use the last ...
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### The power series $\sum_0^\infty2^{-n}z^{2n}$ converges, if

The power series $\sum_0^\infty2^{-n}z^{2n}$ converges, if a) |$z$| $\le$ 2. b) |$z$| $<$ 2. c) |$z$| $\le$ $\sqrt{2}$. d) |$z$| $<$ $\sqrt{2}$. Please anyone give me the answer. I think ...
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### Sum the following $\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)}$

Evaluate: $$\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)}$$ I rewrote the sum as $$\sum_{n=0}^{\infty} \frac {1}{4^{8n-7}(8n-7)} - \sum_{n=0}^{\infty} \frac {1}{4^{8n-3}(8n-3)}$$ Now, I ...
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### Find the sequence $\{c_n\}$ for $c_n = \alpha \cdot c_{n-1} + {\alpha}^{\beta-n}$

Let $\alpha$ and $\beta$ be two given constants, how to find the sequence $\{c_n\}$ for $c_n = \alpha \cdot c_{n-1} + {\alpha}^{\beta-n}$, where $c_0 = {\alpha}^{\beta}$.
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### Testing the convergence of a series [closed]

Does the following series converge? $$\sum_{n=0}^{\infty}\frac{1}{n+3}$$ Please explain with any convergence test you used.