MATH 1620

Final | General | Dabbs

Question 1. Does $\sum_{n=1}^{\infty}(-1)^n\frac{1}{\sqrt{n}}$ converge absolutely, converge conditionally or diverge? Explain your reasoning.

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Question 2. Does $\sum_{n=2}^{\infty} (-1)^n \frac{n^2 - 1}{5n^2 + 3}$ converge absolutely, converge conditionally or diverge? Explain your reasoning.

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Question 3. For what $x$ does the series $\sum_{n=0}^{\infty} \frac{(x-1)^n}{n^32^n}$ converge?

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Question 4. Write the formula for the Taylor series for a function $f(x)$ centered at $x = a$. Find the Taylor series for $f(x) = \sin (x)$ centered at $x = π$

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Question 5. Find the Taylor series for $f(x) = \ln x$ centered at $x = 1$

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Question 6. Find a power series for $(1 + x)^{3/2}$

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Bonus. State and prove the “triple angle identities”(i.e. Give formulas for $\sin(3x)$ and $\cos(3x)$ in terms of $\sin x$ and $\cos x$ alone).

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