Power Series and Taylor Approximation
Cheat sheet.

The key rules, formulas and reminders from all 3 topics, gathered into reference cards.

Key formulas, conditions and traps · Read down each column.

Power Series and Endpoint Tests

Find $R$ from $|x-c|<R$ , then substitute $x=c-R$ and $x=c+R$ into the original series.

The radius does not determine endpoint inclusion. At $R=\infty$ there are no finite endpoints to test.

Differentiate or integrate power series term by term for $|x-c|<R$ . The radius stays the same; endpoints may change.

Track indices and integration constants. Substitution can change the radius in the new variable.

P_n(x)=\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k,\quad |R_n(x)|\le\frac{M|x-a|^{n+1}}{(n+1)!}.

Bound the derivative on the whole interval between center and input. Smoothness alone does not imply equality with a Taylor series.

2 reference blocks

Find $R$ from $|x-c|<R$ , then substitute $x=c-R$ and $x=c+R$ into the original series.

The radius does not determine endpoint inclusion. At $R=\infty$ there are no finite endpoints to test.

2 reference blocks

Differentiate or integrate power series term by term for $|x-c|<R$ . The radius stays the same; endpoints may change.

Track indices and integration constants. Substitution can change the radius in the new variable.

2 reference blocks

P_n(x)=\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k,\quad |R_n(x)|\le\frac{M|x-a|^{n+1}}{(n+1)!}.

Bound the derivative on the whole interval between center and input. Smoothness alone does not imply equality with a Taylor series.