# Tagged Questions

12 views

### Showing that two infinite series converge to the same value

I was preparing for my exam and came across this problem. Show that The series on the left hand side is the power series of $\ln(1+x)$ evaluated at $x=1$. This is what i've done so far. From ...
97 views

48 views

### $\lim_{x\rightarrow {0}^{+}}\sum_{n=1}^{\infty}{(-1)}^{n-1}\frac{1}{n!{x}^{n}}=?$

$$\lim_{x\rightarrow {0}^{+}}\sum_{n=1}^{\infty}{(-1)}^{n-1}\frac{1}{n!{x}^{n}}=?$$ I found this question during my study. In my opinion ,it is not difficult to solve ,but it is interesting. So I ...
47 views
+100

### 'Deriving' the Laplace Transform from the $z$ Transform: Missing a $\Delta t$

Textbooks normally give the following 'derivation' (or justification, if you prefer) of the z-Transform from the Laplace Transform. Let $x(t)$ be a signal defined on $t\geq 0$, and write a discretized ...
66 views

### Radius of convergence of power series which has factorial term

I am trying to find radius of convergence of the following power series: $\sum_{n\geq 1} n^n z^{n!}$ I tried ratio test but it became complicated, I have never seen such radius of convergence problem ...
I rewrote the equation series as $$\frac69 \sum_{n=0}^\infty \left(\frac{-1}{9}\right)^n x^{n+1}$$ And therefore have coefficients of $C_0 = 6/9, C_1 = \left( 6/9 \right) ... 2answers 64 views ### Summation of infinite series If we know the series sum given below converges to a value$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 ... 3answers 27 views ### Finding the sum of a sequence of terms $$1/1(2) - 1/3(2^3) + 1/5(2^5) - 1/7(2^7)$$ This is equal to $$\sum_{n=0}^\infty(1/2)^{2n+1}(-1)^n/(2n+1)$$ Differentiating this leads to: $$\sum_{n=0}^\infty(-1/4)^n$$ Which is equal to$4/5$Thus, ... 3answers 89 views ### Does$x^2+ x^4/(3\cdot4) + x^6/(3\cdot4\cdot5\cdot6) + \cdots$have any compact form? Is there any compact form for the following series $$F_1(x) = x^2+ \frac{x^4}{3\cdot4} + \frac{x^6}{3\cdot4\cdot5\cdot6} + \cdots$$ $$F_2(x) = x+ \frac{x^3}{2\cdot3} + \frac{x^5}{2\cdot3\cdot4\cdot5} ... 0answers 48 views ### Differentiation and integration of a series If I have a power series$$\sum _{k}^{\infty }f(x)$$and I differentiate it I get according to my current knowledge \sum _{k}^{\infty }f(x)',however when I look at a power series defined by$$\sum ... 1answer 30 views ### Analytic Extension: Imaginary Stripe I was always wondering the following: Given a real analytic function there exists a positive radius of convergence for every point. This won't be affected by allowing complex numbers so it extends ... 4answers 61 views ### Why does$\sum_{k=1}^{\infty} \frac {(-1)^{k-1}}{k}$converge conditionally? Why does it converge conditionally? $$\sum_{k=1}^{\infty} \frac {(-1)^{k-1}}{k}$$ 1answer 47 views ### Function whose power series coefficients contain logarithms Is there a function that can be expressed as a power series $$f(x)=\sum_{n=0}^\infty a_n x^n$$ whose coefficients$a_n$are expressions containing$\log n$or something similar? 2answers 52 views ### Product of Infinite Series I am trying to compute the product of 3 infinite series. As such, I need the compact form for the product ... 2answers 51 views ### The Result of Dividing 2 Power Series Is there a way to write a single series for the following division? $$\frac{\displaystyle\sum_{i=0}^{\infty}a_{i}x^{i}}{\displaystyle\sum_{j=0}^{\infty}(j-2)a_{j}x^{j}}$$ Thanks, Radz. 2answers 52 views ### Matrix Inversion Test ( Sum of Matrix series) Friends,I have a set of matrices of dimension$3\times3$called$A_i$. , Following are the given conditions a) each$A_i$is non invertible except$A_0$because their determinant is zero. b) ... 2answers 32 views ### upper bound for the series$S (x) = \sum_{k=1}^{\infty} \frac{(x_n-n)^k}{k!}$from$|x_n -(n+1)|\leq x$. I've been trying to find a tight upper bound for the series $$S (x) = \sum_{k=1}^{\infty} \frac{(x_n-n)^k}{k!}$$ in terms of finite value$x\in \mathbb R$, where: 1-$\{x_n\}$is a sequence of a ... 1answer 44 views ### Rational Series VS Algebraic Series I am reading a paper on combinatorics. It mentions some generating functions are rational series and others are algebraic series. I do not understand the difference, can someone help? EDIT$1$: The ... 2answers 86 views ### Convergence of the series$\sum\limits_{n\geqslant1}(2-x)(2-x^{1/2})(2-x^{1/3})\cdots(2-x^{1/n})$If$x >0$, consider the series$\sum\limits_{n=1}^\infty a_n$, where $$a_{n}=(2-x)(2-x^{1/2})(2-x^{1/3})\cdots(2-x^{1/n}).$$ Is it convergent? This question is in my textbook. I don't know how to ... 1answer 76 views ### Power series difficulty How would I find the region of convergence of the series of$\frac{1}{n^3}(\frac{z+1}{z-1})^n$. I thought about rewriting$\frac{z+1}{z-1}$as$\frac{2}{z-1}+1$but I don't think that helps. Thanks 2answers 28 views ### Radius of convergence query Find the radius of convergence of the series of$\frac{2^n(4z-8)^n}{n}$My answer:$(4z-8)^n=4^n(z-2)^n=2^{2n}(z-2)^n$. Let$c_{n}=\frac{2^{3n}}{n}$. Then$\frac{c_{n}}{c_{n+1}}=\frac{n+1}{2n}$so ... 2answers 29 views ### Find power series representation of$ x/(x^{2}+9)^{2}$I'm not sure how to do it since the entire bottom term is squared. Is there a geometric series I should use? Or differentiation? 2answers 41 views ### how to find convergence and divergence of the series [closed] consider the following two series of complex numbers $$s_1=\sum_1^\infty\frac{i^{n}(2-\sin n)}{2^n.n}$$ $$s_2=\sum_1^\infty\frac{i^n(2-\sin n)}{2^n.n^2}$$ then find whether the above series ... 2answers 71 views ### If a series converges then the power series converges for all z How can I prove that if$\sum \limits_{n=1}^{\infty} c_n$,$c_n\in \mathbb{C}$, converges then$\sum \limits_{n=1}^{\infty} c_n \frac{z^n}{1-z^n}$converges for all z in$\mathbb{C}$with ... 0answers 55 views ### sum of the series$\sum_{k=0}^{\infty}(k+1)(x_n)^k.$Let$x_n$be a sequence of real numbers such that$x_n\in(0,1).$Find the sum of the series$\sum_{k=0}^{\infty}(k+1)(x_n)^k.$My answer is$\frac{(x_n)'}{(1-x_n)^2}.$But the term$(x_n)'$make me ... 0answers 12 views ### How to prove a function$f(t)=\sum_{n=0}^{\infty} a_n t^{2n}$with real coefficients$a_n$is regular in the strip$-\pi/4\le \Im(t) \le \pi/4$? How to prove a function$f(t)=\sum_{n=0}^{\infty} a_n t^{2n}$with real coefficients$a_n$is regular in the strip$-\pi/2\le \Im(t) \le \pi/2$? What will be the sufficient conditions on the real ... 0answers 12 views ### Taylor's expansion of the singular part of an analytic function Assume$f$is analytic on the annulus$R_1<|z-a|<R_2$. Assume$R_1<r<|z-a|$. Define$f_2$by $$f_2(z)=\frac1{2\pi i}\int_{|x-a|=r}\frac{f(x)dx}{x-z}$$$f_2$is analytic on$|z-a|>r$. ... 3answers 53 views ### Radius of$\sum a_n b_n x^n$via radii of$\sum a_n x^n$and$\sum b_n x^n$Series$\sum a_n x^n$and$\sum b_n x^n$have radii of convergence of 1 and 2, respectively. Then radius of convergence R of$\sum a_n b_n x^n$is 2 1$\geq 1 \leq 2$My ... 1answer 45 views ###$\sum_{n=0}^{\infty} a_n x^n$and$\sum_{n=0}^{\infty} a_{n^2} x^n$with different radii of convergence Could you give an example of $$\sum_{n=0}^{\infty} a_n x^n$$ and $$\sum_{n=0}^{\infty} a_{n^2} x^n$$ that have different radii of convergence? 0answers 63 views ### Can there be a power series with interval of convergence$[k, \infty)$? My answer : NO Because Interval of convergence is of the form$(a-R, a+R)$Where$a$is centre of convergence. If there exists a power series with Interval of convergence$[k, \infty) $We ... 2answers 104 views ### Complex series radius convergence How to find the values for which$z$converges,$z\in\mathbb{C}$, in the serie $$\sum_{n=1}^{\infty}\frac{1}{(1+|z|^{2})^{n}}$$ I know I have to use the convergence radius expression, but what I ... 2answers 187 views ### Solution to curious infinite series How exactly does one find a closed form to: $$\sum_{i=0}^{\infty}\left[\frac{1}{i!}\left(\frac{e^2 -1}{2} \right)^i \prod_{j=0}^{i}(x-2j) \right]$$ When expanded it takes on the form $$1 + ... 3answers 77 views ### Can there be more than one power series expansion for a function. I guess the answer is NO, for polynomials. I know that there are more than one series expansion for every function. But I am talking about power series here. All Ideas are appreciated 2answers 129 views ### Convergence of differentiated power series Consider \displaystyle f(z)=\sum_{k=0}^\infty a_k z^k and suppose that \displaystyle\sum_{n=0}^\infty f^{(n)}(0) converges. Prove that \forall z \in \mathbb C, ... 0answers 83 views ### Is there a known evaluation of \sum_{k=1}^{\infty} \frac{1}{k^k}? [duplicate] Is there a known evaluation of$$\sum_{k=1}^{\infty} \frac{1}{k^k}$$Wolfram Alpha says that it converges to \approx 1.29129. 0answers 51 views ### How to power series expand determinants? Say g is a (d\times d) matrix which is given as, g = g_0 + xg_2 + x^2 g_4 .. +x^{d/2 -1}g_{d-2}+ x^{d/2}(g_d + h_d(log (x))) where d is an even number and each g_i is a matrix (same ... 1answer 19 views ### Find the radius of convergence of \sum_{n=0}^{\infty} (3^n + (-5)^n) x^{7n} I have to find the radius of convergence of the series$$\sum_{n=0}^{\infty} (3^n + (-5)^n) x^{7n}$$I know that I will have something like |x^7|<\frac{1}{L}. I tried finding R with ... 1answer 38 views ### Taylor Series expansion and first four terms of 7x^2 e^{-4x} As the series I got$$ \sum_{n=0}^\infty (-1)^n(4x)^n/n! $$which I think is right. However, I am not sure how to get the first four non zero terms. 2answers 32 views ### Meaning of interval of convergence when approximating functions Let's say I have a Taylor series approximation, p(x), of a function f(x) at a:$$ p(x)=\sum_{n=0}^\infty{\frac{f^{(n)}(a)}{n!}(x-a)^n} $$And that this Taylor series has a radius of ... 0answers 47 views ### Sum of a power series I have to find the sum of this series$$\sum_{n=1}^{+\infty}\frac{x^{n-1}}{n!}$$Using integral, I got$$\int\sum_{n=1}^{+\infty}\frac{x^{n-1}}{n!}=\sum_{n=1}^{+\infty}\frac{x^{n}}{n\cdot n!}$$I ... 1answer 24 views ### Convergence of series, using big oh or little oh notation. Let p\in \mathbb{R} and a_n=(e-(1+1/n)^n)^p. For which p will \sum_{n=1}^{\infty} a_n converges? Because of the "additive look" of a_n, I tried to use taylor expansion and big oh, little oh ... 3answers 79 views ### Represent \frac1{(1+x)^2} as a power series Represent \frac1{(1+x)^2} as a power series. if we put \frac1{1+2x+x^2}:$$ \sum _{n=1}^{+\infty}\left(-x-x^2\right)^n $$and if we differentiate \frac{-1}{1+x}:$$ -\sum ... 3answers 58 views ### calculate radius of convergence Let$\{a_n\}_{n=0}^{\infty}$be sequence such that $$a_1 = a_0 = 1$$ $$a_{n+1}=a_n+ a_{n-1}$$ show that the radius of convergence of$\sum\limits_{n=0}^{\infty \:}a_nx^n\$ is ...
While searching for a summation formula I come accross the following equation on wikipedia Equation $$\sum\limits_{k=1}^{n}{k^m z^k}=\left(z\frac{d}{dz}\right)^m\frac{z-z^{n+1}}{1-z}$$ So I tried to ...