Tagged Questions

Convergence of sequences and different modes of convergence.

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Interval of convergence for series with complex numbers

I'm trying to find interval of convergence of this series: $$\sum_{n=1}^{\infty} \frac{7^n(z+2i)^n}{4^n+3^ni}$$ and I should draw a plot which represents the answer, this is what I've got so far: ...
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Convergence problem $\sum \left(1-n\sin\left(\frac{1}{n}\right)\right)$ [closed]

I have to check convergence of: $$\sum_{n=1}^\infty\left(1-n\sin\left(\frac{1}{n}\right)\right).$$ I have no idea but I only check that $\lim \ n\left(1-n\sin\left(\frac{1}{n}\right)\right)=0$.
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Convergence of sequence with $\zeta$ function

Last time I heard interesting question. Unfortunately I do not have idea how to solve it, so I decided to give it here. Let us define sequence $a_n=(\underbrace{\zeta\circ...\circ \zeta}_{n})(\pi)$ ...
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Convergence of $\sum a^{1/x_n}$ for $a$ in $(0,1)$ and $\sum x_n$ a positive convergent series

Let $\sum x_n$ be a convergent series of positive real numbers and $0<a<1$, then is the series $\sum a^{1/{x_n}}$ convergent ? I have only figured out that $\lim a^{1/{x_n}}=0$.
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Show that the Maclaurin series of $f(x)=1/\sqrt{1-x}$ holds for $x\in[0,1/2]$ using the Lagrange remainder theorem I can see that $$f'(x)=\frac12 (1-x)^{-\frac32}\text{ and }f''(x)=\frac12\frac32(1-... 1answer 45 views (Conceptual) Continuity of binary relation \succsim and definition using contour sets Some background information: \succsim is a binary relation that represents preference between two goods. \succsim means "x is at least as good as y." Continuity of this relation is defined to be ... 1answer 80 views Examining a solution of a differential equation without knowing the solution The differential equation is given by$$\dot x=-x \cos x$$with x(0)=x_0\in(0,\frac{\pi}{2}). Now I need to show that for each choice of x_0 the domain of the solution x: I\rightarrow \mathbb{... 1answer 27 views weakly convergence the sequence f_{n}= n. \chi_{[-\frac{1}{n},\frac{1}{n}]} I need to research on the uniform, weak and strong convergence the sequence$$f_{n}= n. \chi_{[-\frac{1}{n},\frac{1}{n}]}$$for n\in \mathbb{N}, in L^{2}(\mathbb{R}) equipped with norm \... 1answer 101 views For what values of a does \int_0^\infty\left(\frac{x^a}{1 + x^2}\right)^4 \, dx converge? I'm learning about convergence/divergence of improper integrals and need help with the following problem: Find for what values of a does the following integrals exists$$(1) \int_0^\infty\...
How do I check divergence of this series? $$\sum_{n=0}^{\infty} \frac{6}{4n-1} - \frac{6}{4n+3}$$ Wolframalpha said it used the comparision test but I don't see what possible smaller sum to use? ...
conditional convergence of $\sum_{n=2}^{\infty} \frac{\cos(n)}{n}$
prove that the series $$\sum_{n=2}^{\infty} \frac{\cos(n)}{n}$$ is conditionally convergent? I tried to prove that it is not absolutely convergent series by trying to prove that \$\sum_{n=2}^{\infty} \...