Infinite Sequences and Series Prep
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What you'll study, topic by topic
Infinite sequences and series
Sample questions with model answers
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Derive the Taylor series for $e^x$ around $x = 0$.
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The Taylor series is $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$.
The full exam-length answer is in the app.
Explain how to determine the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$.
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Use the p-series test; since $p = 2 > 1$, the series converges.
The full exam-length answer is in the app.
Calculate the sum of the infinite series $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...$.
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The sum is $S = 2$ using the formula for an infinite geometric series.
The full exam-length answer is in the app.
Describe the ratio test for convergence of series.
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The ratio test states that if $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1$, the series converges.
The full exam-length answer is in the app.
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