1. What is a power series?
A power series is an infinite sum of powers of \(x\):
\[
\sum_{n=0}^{\infty}a_nx^n.
\]
Many common functions can be represented by power series.
2. What is a partial sum?
A partial sum keeps only finitely many terms:
\[
P_K(x)=\sum_{n=0}^{K}a_nx^n.
\]
As \(K\) increases, \(P_K(x)\) may approach the function \(f(x)\).
3. Approximation error
The error between the function and the partial sum is
\[
E_K(x)=f(x)-P_K(x).
\]
At a specific point \(x_0\),
\[
E_K(x_0)=f(x_0)-P_K(x_0).
\]
4. Convergence interval
A power series usually converges only on a certain interval.
\[
|x|
The number \(R\) is called the radius of convergence.
5. Series for \(e^x\)
\[
e^x
=
1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots
\]
\[
R=\infty,
\qquad
\text{interval }(-\infty,\infty).
\]
6. Series for \(\sin x\)
\[
\sin x
=
x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots
\]
\[
R=\infty,
\qquad
\text{interval }(-\infty,\infty).
\]
7. Series for \(\cos x\)
\[
\cos x
=
1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots
\]
\[
R=\infty,
\qquad
\text{interval }(-\infty,\infty).
\]
8. Series for \(\ln(1+x)\)
\[
\ln(1+x)
=
x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots
\]
\[
R=1,
\qquad
\text{interval }[-1,1).
\]
9. Series for \(\dfrac{1}{1-x}\)
\[
\frac{1}{1-x}
=
1+x+x^2+x^3+\cdots
\]
\[
R=1,
\qquad
\text{interval }(-1,1).
\]
10. Series for \(\arctan x\)
\[
\arctan x
=
x-\frac{x^3}{3}+\frac{x^5}{5}-\cdots
\]
\[
R=1,
\qquad
\text{interval }[-1,1].
\]
11. Why side-by-side plots are useful
The first plot compares \(f(x)\) and \(P_K(x)\). The second plot shows the error:
\[
E_K(x)=f(x)-P_K(x).
\]
When \(K\) increases, the error often gets smaller inside the convergence interval.
13. Common mistakes
- Using too few terms: a low-order polynomial may only approximate well near the center.
- Ignoring the convergence interval: outside the interval, the infinite series may fail.
- Confusing \(P_K(x)\) with \(f(x)\): \(P_K(x)\) is only a finite approximation.
- Judging convergence from one point only: convergence behavior depends on the value of \(x\).
- Forgetting endpoint checks: endpoints such as \(x=-1\) and \(x=1\) need special attention.
