Series Applications

Part 1 of 7 — Approximating Functions

Why Series Matter

Taylor series let us:

1. Approximate transcendental functions with polynomials
2. Evaluate limits that are otherwise indeterminate
3. Compute integrals with no closed-form antiderivative
4. Solve differential equations

Example: Approximating $e$

e = e^1 = sum_{n=0}^{infty}rac{1}{n!} = 1 + 1 + rac{1}{2} + rac{1}{6} + rac{1}{24} + cdots approx 2.71828

Approximation 🎯

Key Takeaways — Part 1

Series provide polynomial approximations to transcendental functions.

Series Applications

Part 2 of 7 — Series Solutions to DEs

Solving $y' = y$ via Power Series

Assume $y = sum c_n x^n$. Then $y' = sum nc_n x^{n-1}$.

$y' = y$: $sum nc_n x^{n-1} = sum c_n x^n$

Matching coefficients: $c_{n+1} = c_n/(n+1)$

With $c_0 = 1$: $c_n = 1/n!$, giving $y = e^x$ ✓

Series & DEs 🎯

Key Takeaways — Part 2

Power series methods solve DEs by matching coefficients.

Series Applications

Part 3 of 7 — Integration via Series

Integrals with No Elementary Antiderivative

int e^{-x^2},dx = int sum rac{(-1)^n x^{2n}}{n!},dx = C + sum rac{(-1)^n x^{2n+1}}{n!(2n+1)}

int rac{sin x}{x},dx = int sum rac{(-1)^n x^{2n}}{(2n+1)!},dx = C + sum rac{(-1)^n x^{2n+1}}{(2n+1)(2n+1)!}

Series Integration 🎯

Key Takeaways — Part 3

Series turn impossible integrals into routine polynomial integrations.

Series Applications

Part 4 of 7 — Finding Coefficients from Derivatives

Connecting Taylor Coefficients and Derivatives

rac{f^{(n)}(a)}{n!} = c_n

So: $f^{(n)}(a) = n! cdot c_n$

This lets you find specific derivatives from a known series without differentiating!

Coefficient-Derivative Connection 🎯

Key Takeaways — Part 4

$f^{(n)}(a) = n! \cdot$ (coefficient of $(x-a)^n$).

Series Applications

Part 5 of 7 — Euler's Formula

The Beautiful Connection

$e^{ix} = cos x + isin x$

This follows from comparing Maclaurin series:

e^{ix} = 1 + ix + rac{(ix)^2}{2!} + rac{(ix)^3}{3!} + cdots

= left(1 - rac{x^2}{2!} + rac{x^4}{4!} - cdots ight) + ileft(x - rac{x^3}{3!} + cdots ight)

$= cos x + isin x$

Euler's Identity

$e^{ipi} + 1 = 0$

Euler Connection 🎯

Key Takeaways — Part 5

Euler's formula connects exponential and trigonometric functions through series.

Series Applications

Part 6 of 7 — Practice Workshop

Mixed Practice 🎯

Workshop Complete!

Series Applications — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Series Applications — Complete! ✅