# Tagged Questions

For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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### Taylor Series for a Function of $3$ Variables

The Taylor expansion of the function $f(x,y)$ is: f(x+u,y+v) \approx f(x,y) + u \frac{\partial f (x,y)}{\partial x}+v \frac{\partial f (x,y)}{\partial y} + uv \frac{\partial^2 f (x,y)...
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### How to prove that the series $\sum\limits_{n=1}^\infty {\sin^2n}$ diverges

I want to use a divergence test to prove that $\lim_{n\to \infty} \sin^2n$ does not converge. So $\sum_{i=1}^\infty \sin^2 n$ diverge. But because $\pi$ is an irrational number. So I cannot use ...
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### Find the next numbers in the sequence.. [on hold]

What is the next number in the sequence: 11, 67, 348, 1071, .... I tried factorizing and taking differences but I can't find common relationship in the sequence.
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### $\succsim$ preorder on X being continuous imply lower contour set closed

$\succsim$ is preorder (i.e. preference relation) on X that is continuous. This implies the lower contour set is closed. Would you please share your 2 cent on my parenthesis explanation (e.g. line ...
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### Understanding how to transform this infinite series into its closed form

On this page, there is an infinite series which the author simplifies "with a little bit of thought" hand-waving. I'm sure the author knew what he was doing but I cannot follow and I am hoping someone ...
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### Right-const function and pointwise/uniform convergence

Let function $f:\mathbb{R} \rightarrow \mathbb{R}$ be right-const iff $\exists_{M \in \mathbb{R}}\forall_{x,y \ge M}f(x)=f(y)$. Consider function sequence $\{f_n\}$ which every term is right-const. ...
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### Generalizing a Telescoping Sum $\sum_{n=1}^\infty \frac{1}{n+k}-\frac{1}{n}$

I was trying to generalize an integral I found yesterday on this website and ran into the following interesting sum: $S_k=\sum_{n=1}^\infty \frac{1}{n+k}-\frac{1}{n}$. I have seen this sum come up a ...
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### Taylor series expansion of function

I have the following statement - If $f$ has a Taylor series expansion about zero with radius $R$ , then $g(x) = \displaystyle f\left(\frac{x-1}{2}\right)$ has a Taylor expansion about $X = 1$ of ...
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### How to prove this series converges ?

I have this series $\sum _{n=0}^{\infty }\:\left(\sqrt[n]{n}-1\right)^n$ Im having truoble to prove that this converges, I've tryind to use the ratio test but it didnt seem to get me to something ...
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### Solution to a first order linear difference equation

The two questions are with respect to the following first order linear difference equation $(Y_{t} - Y_{t-1}) = (1-\lambda) (X_{t-1} - Y_{t-1})$, for $t \geq n$ Also, note that the process ...