Power Series

Part 1 of 7 — Radius & Interval of Convergence

Power Series Form

$sum_{n=0}^{infty} c_n(x-a)^n$

Radius of Convergence $R$

Use the ratio test: lim left|rac{c_{n+1}}{c_n} ight| |x - a| < 1

Radius of Convergence 🎯

Key Takeaways — Part 1

$R$ from ratio test. Always check endpoints separately for the interval.

Power Series

Part 2 of 7 — Checking Endpoints

At each endpoint, substitute and test the resulting series.

sum rac{x^n}{n}, $R = 1$. Interval: check $x = 1$ and $x = -1$.

$x = 1$: $sum 1/n$ → diverges (harmonic)

$x = -1$: $sum (-1)^n/n$ → converges (AST)

Interval of convergence: $[-1, 1)$

Endpoint Checking 🎯

Key Takeaways — Part 2

Always test each endpoint individually. Four possible interval shapes: $(a-R, a+R)$, $[a-R, a+R)$, $(a-R, a+R]$, $[a-R, a+R]$.

Power Series

Part 3 of 7 — Differentiation & Integration of Power Series

Term-by-Term Differentiation

$f(x) = sum c_n x^n implies f'(x) = sum n c_n x^{n-1}$

Same radius $R$ (endpoints may change!)

Term-by-Term Integration

int f(x),dx = C + sum rac{c_n x^{n+1}}{n+1}

Same radius $R$ (endpoints may change!)

Diff/Integration 🎯

Key Takeaways — Part 3

Differentiate and integrate term-by-term. Radius stays the same.

Power Series

Part 4 of 7 — Representing Functions as Power Series

Strategy: Start from Known Series

rac{1}{1-x} = sum x^n, then manipulate!

rac{1}{1+x^2} = sum (-x^2)^n = sum (-1)^n x^{2n}

rac{x}{1-x^3} = x sum (x^3)^n = sum x^{3n+1}

ln(1+x) = int rac{1}{1+x},dx = sum rac{(-1)^n x^{n+1}}{n+1}

Building Power Series 🎯

Key Takeaways — Part 4

Start from $\frac{1}{1-x}$ and use substitution, differentiation, integration.

Power Series

Part 5 of 7 — Non-Zero Centers

Power Series at $x = a$

$sum c_n(x-a)^n$

$R$ found same way. Interval centered at $a$: $(a-R, a+R)$.

Example

sum rac{(x-3)^n}{2^n} = sum left(rac{x-3}{2} ight)^n

Geometric with ratio rac{x-3}{2}. Converges when $|x-3| < 2$, i.e., $1 < x < 5$.

Non-Zero Centers 🎯

Key Takeaways — Part 5

Center at $a$: interval is $(a-R, a+R)$ with endpoint checks.

Power Series

Part 6 of 7 — Practice Workshop

Mixed Practice 🎯

Workshop Complete!

Power Series — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Power Series — Complete! ✅