Finding Taylor or Maclaurin Series for a Function

3. Visual Connection

5. Concept Development

BC-only topic. A Taylor series records every derivative of a function at one center. It is the infinite continuation of the Taylor polynomials from Topic 10.11.

Taylor and Maclaurin series

If \(f\) has derivatives of all orders at \(x=a\), its Taylor series at \(a\) is

At \(a=0\), this becomes the Maclaurin series

From finite polynomial to infinite series

The degree-\(N\) Taylor polynomial is the \(N\)th partial sum:

The Taylor series asks what happens as \(N\to\infty\).

Why the coefficients are unique

If a power series centered at \(a\) represents \(f\), repeated differentiation at \(a\) forces

Therefore a function cannot have two different power-series representations about the same center on an open interval.

A direct-derivative workflow

1. Identify the center \(a\).
2. Compute several derivatives and evaluate them at \(a\).
3. Recognize a sign, parity, factorial, or exponential pattern.
4. Write a formula for \(f^{(n)}(a)\).
5. Substitute it into \(f^{(n)}(a)(x-a)^n/n!\).
6. Display several initial terms to verify the general term.
7. Determine the radius or interval and explain where the series equals \(f\).

Convergence to a number versus representation of f

The Taylor series may converge as a numerical series without converging to \(f(x)\). To write

one must know that the remainders satisfy \(R_N(x)=f(x)-P_N(x)\to0\). A Lagrange bound tending to zero is one common justification.

Smooth does not automatically mean analytic

Having derivatives of every order at a point guarantees that the Taylor series can be formed, but not that it reproduces the function nearby. Representation requires the additional remainder-to-zero conclusion.

Exponential pattern

For \(f(x)=e^{kx}\),

The factorial denominator makes this series converge for every real \(x\).

Sine and cosine cycles

The four-derivative cycles at \(0\) produce

The sine series contains only odd powers; the cosine series contains only even powers. Both converge for all real \(x\).

Logarithm centered at 1

For \(f(x)=\ln x\),

Hence

The left endpoint \(x=0\) gives a divergent harmonic series; the right endpoint \(x=2\) gives the convergent alternating harmonic series.

Reciprocal centered at 1

For \(f(x)=1/x\), \(f^{(n)}(1)=(-1)^n n!\), so

Changing the center

At a nonzero center, preserve powers of \(x-a\). For example, every derivative of \(e^x\) at \(a\) equals \(e^a\), giving

Reading derivatives from a known series

If

then \(f^{(k)}(a)=k!c_k\). Missing powers correspond to zero derivative values at the center.

First terms as an error check

Before finalizing sigma notation, expand the first three or four terms. This catches shifted signs, wrong parity, and factorial-index errors.

AP-style checklist

1. State whether the series is Taylor or Maclaurin and identify its center.
2. Show a valid all-orders derivative pattern.
3. Include the factorial and shifted power in the general term.
4. Check the formula against initial terms.
5. State the radius or interval when requested.
6. Distinguish convergence of the series from equality with the function.

6. Detailed Worked Example and Error Check

Example 1: The exponential series

Because \(f^{(n)}(0)=1\) for \(f(x)=e^x\),

The Ratio Test gives convergence for all real \(x\), and the Lagrange remainder tends to zero.

Example 2: The sine series

The derivative values of \(\sin x\) at zero cycle \(0,1,0,-1\). Therefore

Example 3: The cosine series

The cosine cycle gives

Example 4: Logarithm about x = 1

Using \(f^{(n)}(1)=(-1)^{n-1}(n-1)!\),

The general term is \((-1)^{n-1}(x-1)^n/n\), and the representation interval is \((0,2]\).

Example 5: Reciprocal about x = 1

Since \(f^{(n)}(1)=(-1)^n n!\),

Example 6: Exponential at a shifted center

At \(a=2\), every derivative of \(e^x\) equals \(e^2\). Thus

valid for every real \(x\).

Example 7: Sine centered at pi/2

Writing \(h=x-\pi/2\), \(\sin x=\cos h\). Therefore

This agrees with the direct derivative values \(1,0,-1,0,\ldots\) at \(x=\pi/2\).

Example 8: A scaled exponential found directly

For \(f(x)=e^{2x}\), \(f^{(n)}(0)=2^n\). Hence

Example 9: Recover an arbitrary derivative

Suppose

The coefficient of \((x-2)^k\) is \((k+1)/3^k\), so

Example 10: A polynomial has a terminating Taylor series

For \(f(x)=x^3-2x+1\) centered at \(1\),

All higher derivatives vanish, so the Taylor series terminates and equals the polynomial for every real \(x\).

Common errors

• Writing a finite Taylor polynomial when an infinite series is requested.
• Using \(x^n\) at a nonzero center.
• Forgetting \(n!\) in the coefficient.
• Guessing a general term from one or two terms without checking the derivative pattern.
• Using \(n\) where the sine or cosine pattern requires \(2n\) or \(2n+1\).
• Starting a logarithm series at \(n=0\), which creates division by zero.
• Claiming the Taylor series equals the function merely because both exist.
• Finding a radius but ignoring endpoint behavior.
• Reading \(c_n\) directly as \(f^{(n)}(a)\) instead of multiplying by \(n!\).
• Confusing the interval of convergence with the domain of the original function.
• Dropping the factor \(e^a\) in an exponential series centered at \(a\).