Series & Sequences (Advanced)

Series & Sequences: core idea

• 1. Sum $\sum_{n=1}^{\infty}1/[n(n+1)]$?

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• 2. Maclaurin of $\arctan x$: coefficient of $x^3$?

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• 3. Power series for $\ln(1-x)$: coefficient of $x^2$?

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• 4. Does $\sum \sin n / n^2$ converge absolutely? $1$ yes $0$ no.

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• 5. Alternating error: first unused term bound for $\sum(-1)^{n}/n!$ after $n=3$ term: enter $1/k!$ with $k=$?

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• 6. Cauchy product: coefficient of $x$ in $(e^x)^2=e^{2x}$?

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• 7. Interval of $\sum (-1)^n x^n/n$: left endpoint $x=-1$ converges? $1/0$

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• 8. Find sum $\sum_{n=0}^{\infty}x^n$ at $x=1/2$.

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• 9. Ratio $\lim|a_{n+1}/a_n|$ for $a_n=n^2/3^n$?

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