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4

For $|r|<1$ we have $$|r^n|\cdot|\sin(nx)|\le |r^n|$$ and since $\sum|r^n|$ is a convergent geometric series then the given series is convergent for all $x$ by comparison. Otherwise, and if $x\ne0$ we have $r^n|\sin(nx)|\not\xrightarrow{n\to\infty}0$ hence the series is divergent.

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First, we have \begin{align} \psi'(n) &=\sum_{k=0}^\infty\frac1{(k+n)^2}\\ &=\sum_{k=n}^\infty\frac1{k^2}\tag{1} \end{align} Then \begin{align} \sum_{n=1}^\infty\psi'(n)^2 &=\sum_{n=1}^\infty\sum_{j=n}^\infty\frac1{j^2}\sum_{k=n}^\infty\frac1{k^2}\tag{2}\\ ... 2 Here cancelling a factor n^n works: \frac{n^{n+1/n}}{(n+1/n)^n}=\frac{n^{1/n}}{(1+\frac1{n^2})^n}\to\frac11=1. $$2 Perhaps this way's clearer:$$\frac{n^{n+1/n}}{\left(n+1/n\right)^n}=\frac{\sqrt[n]n}{\left(1+\frac1{n^2}\right)^n}$$1 You have$$f\left(\frac kn\right) \le \int_{k/n}^{(k+1)/n} f(x) \,\mathrm dx \le f\left(\frac{k+1}n\right).$$(This is true simply because the values of the function on the whole interval are between the two values.) If you sum all these inequalities for k=1,\dots,n-1 you get$$s_n \le \int_0^1 f(x) \,\mathrm dx \le t_n.$$This also implies that$$0 \le ...

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By asking what the infinite sum is, would you like to write it down as a decimal (using base 10)? Review of decimals: A decimal $.a_1a_2a_3...$ really is itself an infinite series: $$\sum^{\infty}_{n=1}\frac{a_n}{10^{n}}$$ We are comfortable with such series at least partly because we know exactly how to interpret the accuracy of such an expression: If ...

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There are $8\cdot9^{k-1}$ $k$ digit numbers without a $9$. Even if we take $8\cdot9^{k-1}$ times the reciprocal of the smallest $k$ digit number, we get the overestimate \begin{align} \sum_{j=10^{k-1}}^{10^{k-1}+8\cdot9^{k-1}-1}\frac1j &\le8\left(\frac9{10}\right)^{k-1} \end{align} Add these up for $k\ge1$, we get  \begin{align} ...

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Last Survival is 61 as shown in below matrix 1st 2nd 3rd 4th 5th Final Round 1 1 1 5 13 29 61 2 3 5 13 29 61 3 5 9 21 45 93 4 7 13 29 61 5 9 17 37 77 6 11 21 45 93 7 13 25 53 8 15 29 61 9 17 33 69 10 19 37 77 11 21 41 85 12 23 45 93 13 25 49 14 27 53 15 29 57 16 31 61 17 33 65 18 35 ...

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