7 Power Series and Taylor Series

A progressive guide to representing functions with power and Taylor series, determining convergence, manipulating series, and controlling approximation error.

Power Series as Infinite Polynomials

A power series extends a polynomial to infinitely many terms. Centered at $a$ , it has the form

\sum_{n=0}^{\infty}c_n(x-a)^n=c_0+c_1(x-a)+c_2(x-a)^2+\cdots.

The center determines the expression used for the powers. When $a=0$ , the series is centered at the origin:

\sum_{n=0}^{\infty}c_nx^n.

The partial sum through index $N$ is an ordinary polynomial:

S_N(x)=\sum_{n=0}^{N}c_n(x-a)^n.

As $N$ increases, these polynomials may approach a function on an interval.

The geometric series is the basic model:

\sum_{n=0}^{\infty}r^n=\frac{1}{1-r},\qquad |r|<1.

Replacing $r$ with $x$ gives

1+x+x^2+x^3+\cdots=\frac{1}{1-x},\qquad |x|<1.

For example, rewrite $\frac{1}{1+x^2}$ as $\frac{1}{1-(-x^2)}$ . The geometric series then gives

\frac{1}{1+x^2}=\sum_{n=0}^{\infty}(-1)^nx^{2n}=1-x^2+x^4-x^6+\cdots,\qquad |x|<1.

Takeaway: A power series is an infinite polynomial whose center and coefficients determine its representation and convergence behavior.

Convergence: Radius and Interval

For a power series centered at $a$ , convergence has one of three patterns: it may occur only at $x=a$ , it may occur for every real $x$ , or it may occur inside a finite distance from $a$ . That distance is the , and the full set of convergent real inputs is the .

To find the radius, apply the to consecutive terms. For

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

consider

L=\lim_{n\to\infty}\left|\frac{c_{n+1}(x-a)^{n+1}}{c_n(x-a)^n}\right|.

Absolute convergence occurs when $L<1$ , while divergence occurs when $L>1$ . The resulting inequality usually has the form $|x-a|<R$ .

The endpoints $x=a-R$ and $x=a+R$ require separate tests because the is inconclusive there.

For example, consider

\sum_{n=1}^{\infty}\frac{(x-2)^n}{n3^n}.

The ratio limit is

L=\frac{|x-2|}{3}.

Thus, $|x-2|<3$ , so the possible interval is $(-1,5)$ and the radius is $R=3$ . At $x=-1$ , the series becomes an alternating harmonic-type series and converges. At $x=5$ , it becomes the harmonic series and diverges. Therefore, the is $[-1,5)$ .

Takeaway: Find the interior using a convergence test, then substitute both endpoints into the original series.

Differentiating and Integrating Series

Within the interior of its , a power series can be differentiated and integrated term by term. If

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

has radius $R$ , then for $|x-a|<R$ ,

f'(x)=\sum_{n=1}^{\infty}nc_n(x-a)^{n-1}.

Term-by-term integration gives

\int f(x)\,dx=C+\sum_{n=0}^{\infty}\frac{c_n}{n+1}(x-a)^{n+1}.

Differentiation and integration preserve the same , although endpoint behavior can change.

Starting with

\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n,\qquad |x|<1,

differentiation gives

\frac{1}{(1-x)^2}=\sum_{n=1}^{\infty}nx^{n-1}=1+2x+3x^2+4x^3+\cdots.

Similarly, integrating

\frac{1}{1+x^2}=\sum_{n=0}^{\infty}(-1)^nx^{2n}

produces

\arctan x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1}=x-\frac{x^3}{3}+\frac{x^5}{5}-\cdots, \qquad |x|<1.

Takeaway: Term-by-term calculus is justified inside the convergence interval, but endpoints still need independent attention.

Taylor and Maclaurin Expansions

A centered at $a$ is constructed from the derivatives of a function at $a$ :

\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n.

The first terms are

f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+\cdots.

When $a=0$ , this becomes a :

\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n.

Important include

e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!},\qquad \sin x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!},

and

\cos x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!},\qquad \frac{1}{1-x}=\sum_{n=0}^{\infty}x^n.

The exponential, sine, and cosine series converge for every real $x$ . The geometric series converges only when $|x|<1$ . Another useful expansion is

\ln(1+x)=\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^n}{n}=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots, \qquad -1<x\le 1.

Known series can be adapted by substitution. For $f(x)=\frac{1}{1+x^3}$ , substitute $-x^3$ into the geometric series:

\frac{1}{1+x^3}=\sum_{n=0}^{\infty}(-1)^nx^{3n}=1-x^3+x^6-x^9+\cdots, \qquad |x|<1.

Takeaway: Derivatives generate Taylor coefficients, while known geometric and elementary series provide efficient ways to construct new expansions.

Polynomial Approximation and Error

A is a finite truncation of a :

P_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k.

It is usually most accurate near its center $a$ , and accuracy often improves as the degree $n$ increases.

The error is the remainder

R_n(x)=f(x)-P_n(x).

Taylor's theorem states that for some $c$ between $a$ and $x$ ,

R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}.

If $|f^{(n+1)}(t)|\le M$ on the relevant interval, the is

|R_n(x)|\le\frac{M}{(n+1)!}|x-a|^{n+1}.

For example, a degree- $5$ Maclaurin polynomial for $\sin x$ is

P_5(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}.

At $x=0.2$ ,

P_5(0.2)\approx0.19866933.

Since every derivative of $\sin x$ has absolute value at most $1$ , take $M=1$ :

|R_5(0.2)|\le\frac{0.2^6}{6!}\approx8.9\times10^{-8}.

The polynomial and function have matching derivatives through order $n$ at the center:

P_n^{(k)}(a)=f^{(k)}(a),\qquad 0\le k\le n.

This explains their close local contact, but it does not guarantee equally good accuracy far from $a$ .

Takeaway: Use the polynomial near its center, and use a remainder bound when a guaranteed accuracy is required.

A Practical Problem-Solving Strategy

A reliable workflow keeps convergence and approximation questions separate but connected.

Identify the center. Rewrite the series in powers of $x-a$ .
Find the radius. Apply the or root test.
Test endpoints separately. Substitute each endpoint into the original series.
Use known expansions. Begin with geometric, exponential, sine, cosine, or logarithmic series when appropriate.
Manipulate within the valid interval. Differentiate or integrate term by term only where the original power series converges.
Truncate for approximation. Use a as the finite partial sum.
Check accuracy. Apply the when an error guarantee is needed.

For instance, to approximate a function near $a$ , first select a centered at $a$ , then evaluate it at the desired input. If the input is farther from $a$ , increase the degree or verify the error with Taylor's theorem.

Final takeaway: Power series provide exact representations on suitable domains, while Taylor polynomials provide controlled local approximations. Convergence determines where the representation is valid, and the remainder bound determines how accurate a finite approximation is.