SM122R FLASHCARD 8

Front

What is the Maclaurin series expansion for the function \( e^x \)?

Back

The Maclaurin series expansion for \( e^x \) is \( 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ... \)

Front

How can the function \( f(x) = \frac{1}{1+x} \) be represented as a power series?

Back

The function can be represented as a geometric power series: \( \sum_{n=0}^{\infty} (-1)^n x^n \) for \( |x| < 1 \).

Front

What is the series expansion for \( \cos(2x) \) using the Maclaurin series for \( \cos x \)?

Back

The series expansion for \( \cos(2x) \) is \( 1 - 2x^2 + \frac{2}{3}x^4 - ... \)

Front

What is the Maclaurin series expansion for \( \sin x \)?

Back

The Maclaurin series expansion for \( \sin x \) is \( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ... \)

Front

What is the interval of convergence for the series that represents \( f(x) = e^x \)?

Back

The interval of convergence for the series that represents \( f(x) = e^x \) is \( -\infty < x < \infty \).

Front

What is the general form of a power series?

Back

A power series is generally expressed as \( \sum_{n=0}^{\infty} a_n (x - c)^n \), where \( a_n \) are coefficients and \( c \) is the center of the series.

Front

What is the significance of the radius of convergence in a power series?

Back

The radius of convergence determines the interval around the center \( c \) within which the power series converges to a finite value.

Front

What is the relationship between the Maclaurin series and Taylor series?

Back

The Maclaurin series is a special case of the Taylor series, where the expansion is centered at \( c = 0 \).

Front

How do you find the radius of convergence for a power series?

Back

The radius of convergence can be found using the ratio test or the root test.

Front

What is the Maclaurin series for \( \ln(1+x) \)?

Back

The Maclaurin series for \( \ln(1+x) \) is \( x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ... \) for \( |x| < 1 \).