# Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: recurrence relations, convergence tests, identifying sequences, identifying terms. For questions on finite sums, use the (summation) tag instead.

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### What was Euler's misconception about functions and infinite series?

I just read this on Strichartz' The way of Analysis: [...] Euler - the leading mathematician of the eighteenth - developed all techniques needed for the study of Fourier series, but he never ...
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### Frequency analysis/discrete uniform distribution in multiple choice tests

I may be using the wrong terms in the title but I read that if something is random then each character will occur an equal amounts of times. I read this when reading about the One-Time Pad cipher, ...
2answers
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### representing a recursive difference equation of two variables into one variable equation

suppose the following recursive difference equation ($t$ is time): $$x_t = \frac{a}{1+a}x_{t-1} + \frac{1}{1+a}x_{t+1}$$ where $0<a<1$ is assumed and all values of $a$ at past times are ...
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### Validating a proposition

Proposition: For all $k,n\in\mathbb{Z^+}$ $s.t$ $n\lt4$ $2{n\choose n}+{n\choose n-1}+...+{n\choose k-(n-2)}=2^n$ for $1\le k\le n-1.$ I understand that this proposition is invalid, so are there ...
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### Sequence with a fixed last element Notation

I was trying to write a sequence of two different elements (that always appear in order) with a fixed last element, for an example: $A_1, B_1, A_2, B_2, A_3, B_3, A_4$. I'm not sure which would be the ...
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### Hypothesis testing for equivalence of two arrangements

I have two arrangements(i.e. permutations) of numbers. First one is the target/real arrangement. Second, is the observed arrangement. e.g. Target := 1,2,3,4,5,6,7 Observed := 4,1,7,3,2,5,...
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### Is something wrong in this proof?

Show that if $\sum a_nx^n$ has convergence radio $R$ and $\limsup |a_n| > 0$, then $R\leq 1$. Proof: Suppose that $\sum a_nx^n$ has convergence radio $R$ and $\limsup |a_n|=\alpha > 0$. ...
2answers
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### Proving that $\sum_j x^j$ is differentiable $(-1,1)$

I'm really struggling with understanding how to apply the Weierstrass M-test, and so some hints on this question would be much appreciated: First I want to prove that $\sum_{j = 0}^\infty x^j$ is ...
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### simple undergraduate series quesiton

consider $\displaystyle \sum_{n=1}^\infty (-1)^{n-1}a_n$ where $(a_n)$ is a monotone decreasing sequence of nonnegative numbers with $a_n \rightarrow 0$ by the alternating series test, series ...
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