July 07, 2026 - Calculus 2 - Power Series Multiplication, Calculus, and Taylor Series

A set of practice flashcards focusing on power series multiplication, calculus operations on series, deriving Taylor and Mclloren series, and the relationship between exponential and trigonometric series expressions.

Multiplication of Power Series

A process similar to polynomial multiplication where each term of the second series is multiplied by each term of the first series to find the resulting coefficients for each individual degree.

Calculus on Power Series

Techniques involving term-by-term differentiation and integration of power series representations using the power rule.

Derivative of a Power Series

Applying the power rule to each individual term, which results in the series $\sum n c_n (x-a)^{n-1}$ and causes the index to shift further along.

Integral of a Power Series

Finding the anti-derivative of each term by adding one to the exponent and dividing by the new exponent, typically represented as $C + \sum c_n \frac{(x-a)^{n+1}}{n+1}$.

Radius of Convergence Property

A rule stating that when a power series is differentiated or integrated, the radius of convergence remains the same.

n-th Coefficient Formula ($c_n$)

The formula representing the coefficients of a power series relative to its derivatives, defined as $c_n = \frac{f^{(n)}(a)}{n!}$.

Power Series for $e^x$

The representation of the exponential function centered at zero, given by $\sum_{n=0}^{\infty} \frac{x^n}{n!}$, which converges everywhere.

Power Series for $\sin(x)$

An alternating series consisting only of odd powers and odd factorials, represented as $\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}$.

Power Series for $\cos(x)$

An alternating series consisting only of even powers and even factorials, represented as $\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}$.

Taylor Series

A series representation of a function based on the values of its derivatives at a specific point.

Mclloren Series

A specific case of the Taylor series where the function is centered at zero ($a = 0$).

Ratio Test

A test for convergence where the limit of the absolute value of the ratio $a_{n+1}/a_n$ is evaluated; if the limit is less than one, the series converges.

The Formula (Euler's derivation)

A relationship established using power series where $e^{ix} = \cos(x) + i\sin(x)$, linking exponentials and trigonometric functions.

$$e^{i\pi} = -1$$

Often referred to as the most beautiful formula in mathematics, derived by plugging $\pi$ into the formula $\cos(\pi) + i\sin(\pi)$.