Sequences & Series (Advanced)

Sequences & Series: core idea

• 1. Nested $\sqrt{6+\sqrt{6+\cdots}}$: positive value?

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• 2. Sum $\sum_{k=1}^{n}k\cdot k!$ telescopes to? At $n=4$: value?

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• 3. Generating function sum $x^n$ for $|x|<1$ at $x=1/2$?

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• 4. Fibonacci $F_1=F_2=1$: $F_8$?

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• 5. Prove bound: $a_{n+1}=\sqrt{2+a_n}$, $a_1=\sqrt2$: limit?

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• 6. Weighted sum $\sum_{k=1}^{5}k\cdot2^k$?

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• 7. Binomial $\sum_{k=0}^{4}\binom4k$?

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• 8. Evaluate $\lim_{n\to\infty}\dfrac{n}{n+1}$.

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• 9. Recurrence $a_{n+1}=1/(1+a_n)$, $a_1=1$: $a_3$?

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