Unit 2 Flashcards

Awesome — here’s your Quizlet-style flashcard set 💡
(You can literally copy and paste these directly into Quizlet — one per line as term → definition.)

🟩 Exponential Function — ( e^x )

Q: What is the Taylor (Maclaurin) series for ( e^x )?
A: ( e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} )

Q: What are the first 3 terms of ( e^x )?
A: ( 1 + x + \frac{x^2}{2!} + \dots )

Q: What is the interval of convergence for ( e^x )?
A: ( (-\infty, \infty) )

🟦 Sine Function — ( \sin x )

Q: What is the Taylor (Maclaurin) series for ( \sin x )?
A: ( \sin x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} )

Q: What are the first 3 terms of ( \sin x )?
A: ( x - \frac{x^3}{3!} + \frac{x^5}{5!} + \dots )

Q: What is the interval of convergence for ( \sin x )?
A: ( (-\infty, \infty) )

🟪 Cosine Function — ( \cos x )

Q: What is the Taylor (Maclaurin) series for ( \cos x )?
A: ( \cos x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} )

Q: What are the first 3 terms of ( \cos x )?
A: ( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + \dots )

Q: What is the interval of convergence for ( \cos x )?
A: ( (-\infty, \infty) )

🟨 Geometric Function — ( \frac{1}{1 - x} )

Q: What is the Taylor (Maclaurin) series for ( \frac{1}{1 - x} )?
A: ( \frac{1}{1 - x} = \sum_{n=0}^{\infty} x^n )

Q: What are the first 3 terms of ( \frac{1}{1 - x} )?
A: ( 1 + x + x^2 + \dots )

Q: What is the interval of convergence for ( \frac{1}{1 - x} )?
A: ( |x| < 1 )

🟧 Natural Logarithm — ( \ln(1 + x) )

Q: What is the Taylor (Maclaurin) series for ( \ln(1 + x) )?
A: ( \ln(1 + x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n} )

Q: What are the first 3 terms of ( \ln(1 + x) )?
A: ( x - \frac{x^2}{2} + \frac{x^3}{3} + \dots )

Q: What is the interval of convergence for ( \ln(1 + x) )?
A: ( -1 < x \le 1 )

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