# 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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### The second term of an arithmetic sequence is $13$ and $5^{th}$ term is $31$. What is the $17^{th}$ term of the sequence?

$4.$ Evaluate the following: $a.$ The second term of an arithmetic sequence is $13$ and $5^{th}$ term is $31$. What is the $17^{th}$ term of the sequence? I am trying to do this question (see ...
1answer
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### Fourier coefficients of $\sin x$ from $-π/2\le x\le π/2$

I need to find the fourier coefficients of $\sin x$ from $-π/2\le x\le π/2$, but I have a doubt, is the period $T=\pi$ or $T=2\pi$? And the other doubt is: this is an odd function, right? So, $a_0=0$, ...
1answer
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### Why do $x^{x^{x^{\dots}}}=2$ and $x^{x^{x^{\dots}}}=4$ have the same positive root $\sqrt 2$?

What is $x$ when is satisfies $x^{x^{x^{\dots}}}=2$ ? I am really confused with this; the root is $\sqrt{2}$, but why does the equation $x^{x^{x^{\dots}}}=4$ have the same root?
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### Find the integer part of the sum $S=\sum_{k=1}^{80} \frac{1}{\sqrt k}$

Let $$S=\sum_{k=1}^{80} \frac{1}{\sqrt k}.$$Then I would like to obtain $\lfloor S \rfloor$, the integer part of $S$. I am not able to think how to start question .
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### Limit of partial sums: $\lim_{n\to\infty} \frac1n\sum_{k=1}^n f(k)=0$ if $\lim_{k\rightarrow\infty}f(k)=0$ [duplicate]

I want to argue that $$\lim_{n\rightarrow\infty} \frac{1}{n}\sum_{k=1}^n f(k)=0~~~~~~~ {\rm if}~~~~~ \lim_{k\rightarrow\infty}f(k)=0.$$ This identity does not seem to hold always, but seems to hold ...
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### The sequence $\{a_k\}_{k=1}^\infty$ is increasing. What about $\{b_k\}_{k=1}^\infty$?

Define the $k^\text{th}$ term of the sequence $\{a_k\}_{k=1}^\infty$ of real numbers by $$a_k := k^2 \int_{-\infty}^\infty e^{-kx^2} \, dx.$$ Then $\{a_k\}_{k=1}^\infty$ is an increasing sequence ...
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### Infinite Ordinal Sum

When working with ordinal numbers, would it be correct to say that: $$\sum_{i=0}^{\infty}1 = \omega$$ Or does this simply not make sense? In the ordinals, does the notation $\sum^\infty_{i=0}$ even ...
1answer
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