# 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.

44,415 questions
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### Need an explanation on a certain terminology about converegence range in series.

I found x's possible values, but I am also required to check the edges. What does it mean to check the edges? and why is it necessary for solving the problem?
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### Proving that $a_n$ is not monotonically increasing if $a_n\ge 0$ and $\lim a_n=0$

Had an exam today and there was the following question: Let $a_n$ be a sequence such that for all $n\in \mathbb N$ we have $a_n \geq 0$ and $\lim a_n = 0$. Prove that $a_n$ is not monotonically ...
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### Progressive Roulette Betting

In this problem we discuss a betting strategy known as "progressive betting". Here's the setup: Bets are repeatedly made at a roulette according to the following strategy: All bets are made the same ...
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### how to interchange sum of series formulas?

I have this formula $$\sum_{k=a+1}^{b}\frac{b-k+1}{b-a+1}$$ I want to convert it to $$\frac{1}{b-a+1}\sum_{k=1}^{b-a}k$$ What are the ...
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### A recurrence relation for Fibonacci squared

The sequence $S_n$ is defined as $$S_1=S_2 =1$$ and for $n\ge 2$, $$S_{n+1}=2(A_n + G_n)$$ where $A_n = \frac {S_n+S_{n-1}}{2}$ is the arithmetic mean and $G_n= \sqrt { S_nS_{n-1} }$ is the geometric ...
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### Proving that a sequence in $\ell^2$ is a Cauchy sequence

Let $(x^{(n)})_{n\in\mathbb{N}}, x^{(n)}:= \sum\limits_{i=1}^n \frac{1}{i} e_i$, where $e_i$ is the sequence that is $0$ everywhere but $1$ in the $i^{th}$ element. I would like to prove that this ...
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### If $a_n$ monotonic, $a_n>0$, and $\sum\sqrt {a_na_{n+1}}$ converges, prove $\sum a_n$ converges. [duplicate]

need help solving this problem: Given $a_n$ monotonic, $a_n>0$. prove that if $\sum\sqrt {a_na_{n+1}}$ converges then $\sum a_n$ converges.
### Proving that $\sum_{n=1}^\infty\left(\sqrt[n]{e}-1-\frac 1n\right)$ converges
Problem: Prove that the series $$S=\sum_{n=1}^\infty\left(\sqrt[n]{e}-1-\frac 1n\right)$$ converges. My idea is to change $\sqrt[n]{e}$ to $\sqrt[n]{\left(1+\dfrac 1n\right)^n}$, and then try to ...